強正則圖之研究

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(1)國立交通大學應用數學系碩士論文強正則圖的研究 A Study of Strongly Regular Graph SRG(v, k, λ, µ) base on µ – λ. 研究生：尤貴弘指導老師：黃大原. 教授. 中華民國九十三年六月.

(2) 強正則圖的研究 A Study of Strongly Regular Graph SRG(v, k, λ, µ) base on µ – λ 研究生：尤貴弘. Student: Kuei-Hong You. 指導老師：黃大原教授 Advisor: Tayuan Hwang. 國立交通大學應用數學系碩. 士. 論. 文. A Thesis Submitted to Department of Applied Mathematics College of Science National Chiao Tung University In partial Fulfillment of Requirement For the Degree of Master In Applied Mathematics June 2004 Hsinchu, Taiwan, Republic of China 中華民國九十三年六月.

(3) 強正則圖的研究. 研究生：尤貴弘. 指導老師：黃大原. 教授. 國立交通大學應用數學系. 摘. 要. Friendship定理說明當一個連通圖形滿足任兩點恰有一共同鄰點時，除了三個點的完全圖之外，此圖必不為正則圖。也就是說所有λ=μ= 1 的強正則圖只有K3一個。在這篇文章中，我們從λ的分解來討論一些λ=μ>1 時的正則圖存在的必要條件，並且討論如Ramanujan圖及Symplectic圖等屬於該類圖形的一些性質。除此之外，文章中亦整理了一些μ–λ值不大的正則圖。. 中華民國九十三年六月 i.

(4) A Study of Strongly Regular Graphs SRG(v, k, λ, µ) based on µ − λ Student: Kuei-Hong You. Advisor: Tayuan Huang. Department of Applied Mathematics National Chiao Tung University Hsinchu 30050 Taiwan R.O.C. June 23, 2004. Abstract The friend theorem excludes all strongly regular graphs with µ = λ = 1 except K3 . In this thesis, we study some necessary conditions for strongly regular graphs with µ = λ > 1 based on the decompositions of λ, including the families of Ramanujan graphs and symplectic graphs as examples. A survey of strongly regular graphs with small value of µ − λ were also given in this thesis.. ii.

(5) 誌. 謝. 在這兩年的研究所生活中，感謝我的指導老師黃大原教授對我的包容及在學業及生活上諸多指導及關心，讓我順利的完成論文，並在各個方面都受益良多。也感謝陳秋媛老師的關心與翁志文老師、黃光明老師及傅恆霖老師的教導，讓我接觸到數學中更多不同的領域。其次，要謝謝組合組的所有同學們，抮君、正傑、昭芳、喻培、宏嘉、棨丰、嘉文、致維、建緯、啟賢、文祥，特別是致維的大學同學們，讓我開心快樂地度過這兩年。感謝我大學時代的同學德筌總是能在我困惑的時候給予適當的意見及建議。還有要感謝祐寧、曲敏、宜諴等學妹們大力協助我的論文的完成，有了妳們讓我輕鬆多了。最後感謝所有幫助過我的所有人，希望我將來也能成為幫助別人的人。. iii.

(6) Contents Abstract (In Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract (In English) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction. 1. 2 Preliminaries 2.1 Matrix interpretations of SRG and BIBD . . . . . 2.2 A technique in terms of local eigenvalues . . . . . 2.3 Some feasible parameters of SRG based on µ − λ 2.4 Some families of SRG with certain properties . . . 3 The 3.1 3.2 3.3. i ii iii. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 3 4 8 10 11. Friendship Property and Strongly Regular graphs 17 A review of friendship theorem . . . . . . . . . . . . . . . . . . . . . 17 SRG with µ = λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Symplectic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. 4 Bent Functions, Ramanujan graphs and SRG 27 4.1 SRG associated with bent functions . . . . . . . . . . . . . . . . . . . 27 4.2 SRG which are Ramanujan graphs . . . . . . . . . . . . . . . . . . . 31 5 SRG with Small µ − λ 33 5.1 SRG with µ = λ + 1 and conference graphs . . . . . . . . . . . . . . . 33 5.2 Symmetric 2-designs from SRG with µ = λ, λ + 2 . . . . . . . . . . . 34 5.3 SRG associated with quasi-symmetric designs . . . . . . . . . . . . . 36 Bibliography A A table of SRG A.1 µ = λ . . . A.2 µ = λ + 1 . A.3 µ = λ + 2 .. 37 on . . . . . .. at . . . . . .. most 280 . . . . . . . . . . . . . . . . . .. vertices 41 . . . . . . . . . . . . . . . . . . . . . . 41 . . . . . . . . . . . . . . . . . . . . . . 43 . . . . . . . . . . . . . . . . . . . . . . 46. iv.

(7) A.4 Unique existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.5 SRG but not Ramanujan graph . . . . . . . . . . . . . . . . . . . . . 50. v.

(8) Chapter 1. Introduction. The theory of designs concerns itself with questions about subsets of a set possessing a high degree of regularity. By contrast, the large and amorphous area called ”graph theory” is mainly concerned with questions about general relations on a set. There are some places where the two theories have interacted fruitfully. The unifying theme is provided by a class of graphs, the strongly regular graphs, introduced by Bose (1963), whose definition reflects the symmetry inherent in t-designs. There are some easy examples that the block graph of quasi-symmetric designs, and line graphs of 2-(v, k, 1) designs are strongly regular graphs. In addition to design theory, strongly regular graphs occur in many areas of computer science like digital logic, network security, and telecommunication networks. Since 0 and 1 are the only messages computer can recognized, Boolean function is used frequently. Spectral techniques have been widely used since the 70s in logic synthesis, testing, function classification, and other applications in logic circuits. Moreover, several authors have analyzed the Walsh spectrum of Boolean functions, and found links between properties of the spectrum and certain computational questions related to the functions. There are many useful tools for analyzing strongly regular graphs since the existence of strongly regular graphs are not necessarily determined by their parameters. Some necessary conditions over v, k, λ, µ of SRG(v, k, λ, µ) are given in Chapter 2 including Krein condition and Seidel’s absolute bound. A few families, such as T (n), L2 (n), Payley graph, etc., of SRG are also included in Chapter 2. Besides, the technique of matrix is another tool which associated algebra theory to graph theory. In analyzing a problem, decomposing a question into small pieces of easier questions is used frequently. The technique of local eigenvalue is a method discussing an introduced subgraph, with smaller order, of a strongly regular graph with known parameters, or constructing a graph from its introduced subgraphs. A common. 1.

(9) feature among SRG, BIBD and BGW matrices is studied in Section 2.5. Viewing a vertex as a person in a party, since we only discussing simple graphs, the edge with no direction means the relation between any pair of persons is symmetric. In Chapter 3, we review the history of friendship theorem and generalize it as a SRG with µ = λ. We discuss the decomposition of λ, use the divisors of λ to find the formula of v and k which is a feasible parameters set, and give some feasible conditions from it. Finally, we present a family of SRG(v, k, λ, λ), symplectic graph, to end this section. As we consider a Cayley graph associated a Boolean function, bent function introduced in Section 4.1, a special case of Boolean function is related to a class of strongly regular graph with µ = λ. Bent functions is often used to build the S-box in conventional encryption of network security. In the telecommunication network, √ are all Ramanujan graphs, introduced strongly regular graphs with |µ − λ| ≤ 23k−4 k−1 in Section 4.2. Ramanujan graph is very interesting in telecommunication network because of its small absolute value, which implies small diameter, of the next to the largest eigenvalue. It means that a Ramanujan graph can be used to construct a good communication network which spreads information fast and costs less. A survey of strongly regular graphs with small |µ − λ| will be given in Chapter 5, including the family of Moore graphs with (λ, µ) = (0, 1) and the family of conference graphs with µ−λ = 1 in general. In addition to symplectic graphsand bent functions, mentioned in Chapters 3 and 4, symmetric 2-designs also provide some strongly regular graphs with µ − λ = 0 or 2, some relations between them can be found in Section 5.2.. 2.

(10) Chapter 2. Preliminaries In this chapter, some basic properties of SRG and designs are given in Section 2.1, their matrix representations together with spectrums will be considered too. From these properties, some useful necessary conditions will be represented. In Section 2.2, we give some techniques which are useful in computing the eigenvalues of adjacency matrices and we can see some relations between the adjacency matrices of the first and second subconstituents. Definition 1. Let Γ be a k-regular graph of order v with the following properties that each pair of adjacent vertices has λ common neighbors and each pair of nonadjacent vertices has µ common neighbors. Then Γ is said to be strongly regular with parameters (v, k, λ, µ). Theorem 2.0.1 ([19]pp.218). Assume that Γ is a SRG(v, k, λ, µ). 1. The complement graph Γ of Γ is also a strongly regular graph with parameters (v, k, λ, µ) where k = (v − 1) − k, λ = (v − 2) − 2k + µ, µ = (v − 2) − (k − 1) − (k − 1) + λ. 2. k(k − λ − 1) = (v − k − 1)µ. Parameters Γ v Γ v. k. λ. v − k − 1 v − 2k − 2 + µ. µ v − 2k + λ. Eigenvalues √ 2. (λ−µ)±. √ (µ−λ−2)±. (λ−µ) +4(k−µ) 2 (µ−λ−2)2 +4(k−λ−1) 2. The second property can be obtained by counting the number of edges between Γ1 (x) and Γ2 (x) for arbitrary vertex x in two ways where Γj (x) is defined as the set of vertex u satisfying d(x, u) = j. For the connectivity of Γ, a SRG(v, k, λ, µ), there are some statements are equivalent: Γ is not connected, µ = 0, λ = k − 1, and Γ = mKk+1 for some m > 1.. 3.

(11) Definition 2. A connected graph Γ is called distance-regular if there are integers bi , ci , i = 0, 1, ..., d such that for any distinct vertices x, y with d(x, y) = i , then Γi−1 (x) ∩ Γ1 (y) = bi and Γi+1 (x) ∩ Γ1 (y) = ci .. 2.1. Matrix interpretations of SRG and BIBD. Since the complete graphs are the trivial strongly regular graphs, we only consider those not complete and whose complement is also connected. Suppose A is the adjacency matrix of a connected strongly regular graph Γ. The entry uv of A2 is the number of walks of length 2 from u to v. Therefore, from the definition of strongly regular graph we have A2 = kI + λA + µ(J − I − A) which can be rewritten as A2 − (λ − µ)A − (k − µ)I = µJ. Since Γ is k-regular, k is an eigenvalue with all one vector as its eigenvector, other eigenvalues of A can be found explicitly. The first part of the following theorem yields a powerful feasibility condition. Given a parameter set we can compute mθ and mτ using these formulas. If the results are not integers, then there is no such a strongly regular graph with these parameters. In practice this is a very useful condition, called the Integrality Condition, as we shall see in Theorem 3.2.1. The classical application of this idea is to determine the possible valencies for a Moore graph with diameter 2. The third part is another bound called absolute bound. Two methods are known to derive the final part of the following theorem, called Krein bound. The first one, consider a strongly regular graph as a two-class association scheme, see details in [27, pp.237-238]. The second one is more elementary, which uses Cauchy-Schawrtz inequality and the discriminant of a quadratic polynomial to get these two inequalities, but seems not easy to generalize to distance-regular graphs in general. Theorem 2.1.1 ([19], pp.220-221). Suppose Γ is a SRG(v, k, λ, µ) with Spec(Γ) = (k 1 , θmθ , τ mτ ), then √ √ (λ − µ) − ∆ (λ − µ) + ∆ ,τ= , with multiplicities 1. θ = 2 2 1 2k + (v − 1)(λ − µ) √ mθ = ((v − 1) − )and 2 ∆ 1 2k + (v − 1)(λ − µ) √ mτ = ((v − 1) + ) 2 ∆ respectively, where ∆ = (θ − τ )2 = (λ − µ)2 + 4(k − λ); 4.

(12) 2. λ = k + θ + τ + θτ , µ = k + θτ and mθ mτ (θ − τ )2 = vkk; 3. (the absolute bound) v ≤ 21 mθ (mθ + 3) and v ≤ 12 mτ (mτ + 3); 4. (the Krein bound). θτ 2 − 2θ2 τ − θ2 − kθ + kτ 2 + 2kτ ≥ 0 and θ2 τ − 2θτ 2 − τ 2 − kτ + kθ2 + 2kθ ≥ 0. If the first inequality is tight, then k ≥ mθ , and if the second is tight, then k ≥ mτ . If either of the inequalities is tight, then one of the following is true: (a) Γ is the 5-cycle C5 . (b) Either Γ or Γ has all its induced subgraphs over Γ1 (x) empty, and all its induced subgraphs over Γ2 (x) strongly regular. (c) All subconstituents of Γ are strongly regular. Proof. 1. Since Γ is k-regular, the all one vector is the eigenvector of eigenvalue k and other eigenvectors are orthogonal to it. From A2 − (λ − µ)A − (k − µ)I = µJ, the remaining eigenvalues θ and τ which are represented as the theorem are the roots of the quadratic equation x2 − (λ − µ)x − (k − µ) = 0. Because Γ is connected and the sum of the eigenvalues equals trace(A) = 0. It follows that +k and mτ = (v−1)θ+k . the corresponding multiplicities are 1, mθ = − (v−1)τ θ−τ θ−τ √ √ (λ−µ)+ ∆ (λ−µ)− ∆ and τ = , 2 2 √ ). 1) + 2k+(v−1)(λ−µ) ∆. Applying θ = mτ = 12 ((v −. √ mθ = 21 ((v − 1) − 2k+(v−1)(λ−µ) ) and ∆. 2. We only prove the last equality. mθ mτ = (−. −(v − 1)2 θτ − (v − 1)(θ + τ )k − k 2 (v − 1)τ + k (v − 1)θ + k )( )= θ−τ θ−τ (θ − τ )2. Replace θτ by µ − k, θ + τ by λ − µ and k by v − 1 − k, then mθ mτ (θ − τ )2 = −(v − 1)2 θτ − (v − 1)(θ + τ )k − k 2 = (v − 1)2 (k − µ) − (v − 1)(λ − µ)k − k(v − 1 − k) = (v − 1)((v − 1 − k)k − (v − 1 − k)µ + k(k − λ − 1)) + kk. Because (v−1−k)µ = k(k−λ−1), hence mθ mτ (θ−τ )2 = (v−1)kk+kk = vkk. 5.

(13) There is a theorem showing us that we can easily determine whether a graph is strongly regular or not by the number of distinct eigenvalues and their multiplicities, since it is a necessary and sufficient condition for an existent graph. Theorem 2.1.2. A graph is a strongly regular graph if and only if it has exactly three eigenvalues. Proof. Because A has exactly three distinct eigenvalues, the minimal polynomial of A is f (x) = (x−k)(x−θ)(x−τ ) = (x−k)g(x). Therefore, f (A) = (A−kI)g(A) = 0 and Ag(A) = kg(A). Hence, g(A) = tJ for some t, and J is a linear combination of A2 , A, and I. Definition 3. Let v 0 , k 0 and λ0 be positive integers such that v 0 > k 0 ≥ 2. A (v 0 , k 0 , λ0 )-balanced incomplete block design (which we abbreviate to (v 0 , k 0 , λ0 )-BIBD), which is also called a 2-(v 0 , k 0 , λ0 ) design, is a pair (X, B) such that the following properties are satisfied: 1. X is a set of v 0 elements called points, 2. B is a collection of subsets of X called blocks, 3. each block contains exactly k 0 points, and 4. every pair of distinct points is contained in exactly λ0 blocks. It is often convenient to represent a BIBD by means of an incidence matrix. Let X = {x1 , x2 , ..., xv0 } and B = {B1 , B2 , ..., B( b }. The incidence matrix of (X, B) is the 1 if xi ∈ Bj v 0 × b 0-1 matrix M = (mij ) where mij = 0 if xi 6∈ Bj From the definition, we have JM = kJ and M M T = rI + λ(J − I) for any incidence matrix M of a 2-design. A 2-design in which b = v is called symmetric. Here are some properties of symmetric 2-designs. Theorem 2.1.3 ([32], pp.12). Let Π = (X, B) be a 2-(v 0 , k 0 , λ0 ) design. The followings are equivalent. 1. |Bi ∩ Bj | = λ0 for any pair Bi , Bj ∈ B; 2. Π is a symmetric design.. 6.

(14) From the above theorem, a symmetric 2-design has the property that the intersection of each pair of distinct blocks has unique cardinality. If the number of cardinalities becomes two, then this 2-design is called a quasi-symmetric design. Let x < y be the two cardinalities of block intersection in a quasi-symmetric design. The block graph of a quasi-symmetric design has as vertices the blocks, two vertices adjacent if they intersect y points. As we know, a connected graph is a strongly regular graph if and only if its adjacency matrix has exactly three distinct eigenvalues. By the property, a block graph of a quasi-symmetric design is a strongly regular graph and we will represent the parameters of SRG by its eigenvalues, see details in Section 5.3. Definition 4. Let Π = (X, B) be a 2-(v 0 , k 0 , λ0 ) design. Π is called an affine resolvable design if the following properties are satisfied: 1. There is a partition {π1 , π2 , ..., πr } of B such that each πi is a set of disjoint blocks whose union is X.(resolvable property) 2. b = v 0 + k 0 + λ0 − 1 where |B| = b. Since any two blocks from different πi of an affine resolvable design intersect in 2 2 exactly kv points [32, pp.36], it is a quasi-symmetric design with x = 0 and y = kv . In order to associated algebraic theory to the area of graph theory, we transfer a graph to an adjacency matrix or an incidence matrix. Similarly, design theory can be associated to algebraic theory by the incidence matrix of a design. The previous two classes of matrix are 0-1 matrices; there is another class of matrix whose entries are not just 0 and 1 but the elements of a finite group, called balanced generalized weighing matrix, having similar behavior as the adjacency matrix of a strongly regular graph and the incidence matrix of a symmetric 2-design. Definition 5. Let (G, ·) be a finite group, W = (αij )v×v a matrix over G ∪ {0}. If each row of W contains exactly k nonzero entries and, for distinct i, h ∈ {1, 2, ..., v}, −1 λ copies of the multi-set {αhj αij : 1 ≤ j ≤ v, αij 6= 0, αhj 6= 0} contains exactly |G| each element of G. Then W is called a balanced generalized weighing matrix with parameters (v, k, λ) over G or a BGW (v, k, λ). d+1. Most of the known BGW belong to the family ( q q−1−1 , q d , q d − q d−1 ) over a cyclic group G, where q is a prime power, d is a positive integer, and the order of G divides q − 1. Suppose R is the group ring of a finite group G over the rationals. −1 ∗ T ∗ ∗ W ∗ = (αij ) , αij if αij ∈ G and αij = αij = 0 if αij = 0. Then W is a BGW (v, k, λ) if and only if λ G)(J − I) W W ∗ = (ke)I + ( |G| 7.

(15) where e is the identity of G, G stands for the sum of all elements of G in R, I is the identity matrix, and J is the matrix with all entries equal to e. The form is similar to the quadratic form of N N T = kI + λ(J − I) where N is an incidence matrix of a symmetric 2-(v, k, λ) design. If N is also a symmetric matrix with all diagonal entries equal to 0, then N is an adjacency matrix of SRG(v, k, λ, λ). Because A2 = kI + λA + µ(J − I − A) for any adjacency matrix A of a strongly regular graph.. 2.2. A technique in terms of local eigenvalues. If we have constructed a graph and want to know whether it is strongly regular or not, the previous section gives a sufficient and necessary condition by computing its eigenvalues. But given a feasible parameter set, it is still difficult to know the existence of the corresponding graph. There is much algebraic theory useful to analysis the adjacency matrix of graph. C. Godsil and G. Royle [19] discuss the properties of the eigenvalues of two special induced subgraphs, whose vertex sets are the vertices at distance one or two from an arbitrary vertex respectively, and use it to prove the uniqueness of Clebsch graph.   0 1T 0   Definition 6. Let A =  1 A1 B T  be the adjacency matrix of a strongly regular 0 B A2 graph Γ with a partition {u} ∪ Γ1 (u) ∪ Γ2 (u) of V (Γ). 1. A1 and A2 are called the first and second subconstituents of graph Γ relative to u respectively. 2. If an eigenvalues of Ai , i = 1, 2 is not an eigenvalue of A and orthogonal to the all one vector, then it is called a local eigenvalue. In discussing the subconstituents of a strongly regular graph, there is an interesting result that 5-cycle is the only connected strong regular graph whose complement is connected, A1 is empty, and A2 is complete. And the Clebsch graph(see details in Section 2.4) provides an example where A1 is empty and A2 is Petersen graph. These two graphs approach the tight Krein bound as Theorem 2.1.1. in previous section.. 8.

(16) Definition 7. A partition π = {C1 , C2 , ..., Cr } of V (Γ) is called an equitable partition if the number of neighbors in Ci of a vertex u in Cj is a constant cij , which is independent of u. The first and second subconstituents of a strongly regular graph are regular graphs and the edges joining any two distinct cells forms a semiregular bipartite graph. The partition π = {u} ∪ Γ1 (u) ∪ Γ2 (x) of V (Γ) is an equitable partition. For any adjacency matrix A as following, each diagonal cell is an adjacency matrix of an induced subgraph.   A1 B1 B2   A =  B3 A2 B4  B5 B6 A3 If the rowsums of each submatrix of A are the same, i.e., π is also an equitable partition, then we have a 3 × 3 matrix   rowsumof A1 rowsumof B1 rowsumof B2   A/π =  rowsumof B3 rowsumof A2 rowsumof B4  rowsumof B5 rowsumof B6 rowsumof A3 The three eigenvalues of A/π are identical with those of A, and we can see that every strongly regular graph has an equitable partition which make the rowsums of each submatrix are the same and   0 k 0   A/π =  1 λ k − λ − 1  0 µ k−µ Here we introduce a technique used to compute the eigenvalues of adjacency matrices, and can be used in proving following lemmas. Suppose there is a quadratic equation of matrix A. Assume an eigenvalue of A is θ, and Ax = θx, x 6= 0. Multiply eigenvector x to both sides of the equation. Then, the equation can be transferred to an equation of θ. For example, A2 + aA + bI = B T B, and θ1 , θ2 are the roots of θ2 + aθ + b = 0 with θ1 > θ2 ; 1. if Bx = 0, then (A2 + aA + bI)x = B T Bx = 0, θ2 + aθ + b = 0, hence θ1 + θ2 = −a, θ1 θ2 = b. 2. if Bx 6= 0, then (A2 + aA + bI)x = B T Bx, θ2 + aθ + b > 0 and we have θ2 < θ < θ1 .. 9.

(17) Lemma 2.2.1 ([19], pp.228). Let Γ be strongly regular with Spec(Γ) = (k 1 , θmθ , τ mτ ), A1 and A2 be the first and second subconstituents respectively. 1. suppose A1 x = θ1 x with 1T x = 0, then θ1 ∈ {θ, τ } whenever Bx = 0, or otherwise τ < θ1 < θ; 2. suppose A2 y = θ2 y with 1T y = 0, then θ2 ∈ {θ, τ } whenever B T y = 0, or otherwise τ < θ2 < θ. 3. σ is a local eigenvalue of one subconstituent of Γ if and only if λ − µ − σ is a local eigenvalue of the other, with equal multiplicities. Proof. We only prove the third part. A1 x = σ1 x, applying BA1 + A2 B = (λ − µ)B + µJ, A2 (Bx) = (λ − µ − σ2 )(Bx); A2 y = σ2 y, A1 (B T y) = (λ − µ − σ2 )(B T y); the dimension of σ1 -eigenspace is not less than the dimension of (λ − µ − σ1 )-eigenspace; and vice versa.. 2.3. Some feasible parameters of SRG based on µ − λ. It is known [11] that there are only finite many feasible parameter sets (v, k, λ, λ) for a given λ. We also give some theorems in Section 3.2 to sieve the possible parameter sets of this class of strongly regular graphs from the value of λ. Berlekamp and van Lint proved that there are only finitely many feasible parameter sets (v, k, λ, λ + 1) for each λ. Strongly regular graphs with µ−λ = 2 corresponding to symmetric 2-(v 0 , k+1, µ) designs with µ ≥ 2 that have a polarity with all points absolute (see details in Section 5.2). The projective planes provide an infinitely family of symmetric 2-(v 0 , k + 1, µ) designs with (v 0 , k 0 , µ) = (k 2 + k + 1, k + 1, 1), but only finitely many symmetric 2-(v 0 , k + 1, µ) designs are known for each µ ≥ 2. Thus it is currently unknown whether there are infinitely many SRG(v, k, µ − 2, µ) for a given µ ≥ 2. Strongly regular graphs with µ = 1, λ = 0 are discussed by Kantor [24] and N. Biggs [8] respectively. Lemma 2.3.1 ([24]). 1. For each λ 6= 3, the parameter set (v, k, λ, 1) is feasible if and only if k is one of a finite list of values. 2. The parameter set (v, k, 3, 1) is feasible if and only if k = r2 where r ≥ 4 is even. Lemma 2.3.2 ([8], pp.102). 10.

(18) 1. The parameter set (v, k, 0, 2) is feasible if and only if k = r2 + 1 where r 6= 0(mod 4), r ≥ 2. 2. The parameter set (v, k, 0, 4) is feasible if and only if k = r2 . 3. The parameter set (v, k, 0, 6) is feasible if and only if k = r2 − 3 where r ≥ 3 and r 6= 0 (mod 4). 4. For µ 6∈ 2, 4, 6, the parameter set (v, k, 0, µ) is feasible if and only if k is one of a finite list of values. Elzinga [17] proved the following theorem. Theorem 2.3.3 ([17]). For fixed λ and µ, there are only finite many feasible parameter sets (v, k, λ, µ), unless λ and µ satisfy one of the following three relations: 1. (µ − λ)2 = 4µ; 2. µ − λ = 2; 3. (µ − λ)2 − 2(µ − λ) = 4µ. The parameter sets (v, k, λ, µ) for which λ and µ satisfy one of the above three equations are summarized in the following table for some integer t and nonnegative integer r. Case 1. 2. 3.. v. k. λ. µ. (r2 +r−t)(r2 −r−t) t2 (r 2 +µ)2 −r2 µ r 2 (r 2 −1) t(t+1). r2. t2 + 2t. t2. r2 + µ − 1. µ−2. µ. r2 + t. t2 + 3t t2 + t. mθ r(r 2 +r−t)(r−t−1) 2t2 (r 2 +r+µ)(r 2 +µ−1) 2µ (r2 +t)(r−t−1)(r+1) 2t(t+1). Note that µ − λ is even in either of the above cases. If one of the relations in the previous theorem holds, then there may be infinitely many feasible (v, k, λ, µ) .. 2.4. Some families of SRG with certain properties. A few families of SRG will be presented in this section. Petersen graph, Clebsch graph and Schläfli graph, are three graphs whose second subconstituent is the previous one and we only prove the uniqueness of Petersen graph which helps to construct the figure of the last two. L2 (n), the square lattice graph, and T (n), the triangular graph, are two families of unique SRG with their parameters as n is large enough. Gewirtz graph and Payley graph are two kinds of strongly regular graph which can be constructed from group theory. Finally, a family of strongly regular graph from 11.

(19) 2-(v 0 , k 0 , 1) design is presented. There are several kinds of graphs from symmetric 2-designs and quasi-symmetric designs will be introduced in Section 5. The Payley graph P (q) is defined on the finite field GF (q) where q ≡ 1(mod4), such that x, y ∈ GF (q) are adjacent if and only if x − y = c for some nonzero square c in GF (q). Theorem 2.4.1 ([19], pp.221). A Payley graph P (q) is a SRG(q, √ q−1 √ q−1 q − 1 1 −1 + q 2 −1 − q 2 q−1 ) and Spec(P (q)) = ( , ). , 4 2 2 2. q−1 q−5 , , 2 4. The family of Payley graph is another family of Cayley graph which will be introduced in Section 4.1. The Cayley set is the collection of x which can be represented as a square. Since −1 is a square in GF (q), the graph is undirected. A Payley graph is a conference graph, which has the parameters set as the above theorem. Indeed, Payley graphs are self complementary. Here are some figures of Payley graph of small orders. Definition 8. The Gewirtz graph is defined on {∞} ∪ P ∪ Q, where P is the set of Sylow 3-subgroups of the alternating group A6 , and Q the set of involutions in A6 . Join ∞ to all vertices in P ; join p ∈ P to q ∈ Q whenever q −1 pq = p; join q1 , q2 ∈ Q whenever q1 q2 has order 4. Combinatorially, we may identify p ∈ P with a pair of disjoint 3-subsets of {1, ..., 6} (its orbits). Then typical edges of the second and the third types join {{1, 2, 3}, {4, 5, 6}} to (12)(45), and (12)(34) to (23)(56) respectively. Definition 9. Let Y be a subset of V (Γ), the vertex set of a graph Γ. The graph switching Γ with respect to Y arises from Γ by changing all the edges between Y and V (Γ) \ Y to non-edges, and all the non-edges between Y and V (Γ) \ Y to edges. Definition 10. Let [n] denote the set {1, 2, ..., n}. 1. The Petersen graph is defined on [5] , such that A, B ∈ 2 and only if |A ∩ B| = 0.. [5] 2. . are adjacent if. 2. The Clebsch graph is defined on {A : A ∈ 2[5] , |A| = 0 or 2 or 4}. A and B are adjacent if and only if |A ∪ B − A ∩ B| = 4. [n] 3. The triangular graph T (n) is defined on [n] , such that A, B ∈ are ad2 2 jacent if and only if |A ∩ B| = 1. Any vertex neighborhood subgraph of T (n) is isomorphic to the product Kn × K2 , where Kn is the complete graph on n vertices. 12.

(20) Figure 2.1: Petersen graph. Figure 2.2: Clebcsh graph. Figure 2.3: Shrikhande graph 4. The square lattice graph L2 (n) is defined on [n] × [n], and two vertices are adjacent if and only if they agree in one coordinate. 5. The Shrikhande graph is obtained from L2 (4) by switching with respect to the set of vertices {(1, 1), (2, 2), (3, 3), (4, 4)}. It is strongly regular with the same parameters as L2 (4). 6. The Schl¨ af li graph is obtained from T (8) by deleting {1, 2} and switching with respect to the set Ω = {{1, i}, {2, i} : i = 3, ..., 8}. i.e., it has the vertex set [n] − {{1, 2}}, and any two vertices in Ω (or in ΩC ) are adjacent if and only 2 if they are not disjoint, on the other hand, one vertex in Ω and another vertex in ΩC are adjacent if and only if they are disjoint. 7. The cocktail party graph CP (n) is the complement of nK2 . It models a cocktail party made up of n couples, at which each participant speaks to everybody except her/his partner. 8. The three Chang graphs are obtained by switching T (8) with respect to (a) four disjoint edges; (b) an octagon; (c) the disjoint union of a pentagon and a triangle. The vertex neighborhood subgraphs of the three Chang graphs are no longer strongly regular graphs, but those of the first Chang graph still possess highly symmetric structure. The first Chang graph is obtained from T (8) by switching with 13.

(21) respect to {{1, 2}, {3, 4}, {5, 6}, {7, 8}}, and independent set. Each maximal clique if the vertex neighborhood subgraph of {1, 2} consists of 4 vertices, and the intersection of any two of them is either empty or a singleton. If any two distinct maximal cliques are called adjacent whenever they are not adjacent, then the associated graph on the set of all maximal clique is isomorphic to the multipartite graph K2,2,2 . Some interesting relations are among T (5), Petersen graph, Clebsch graph, and Schläfli graph, related to the work of Cameron et al [14], and Noda [29]: 1. The vertex neighborhood subgraph Γ(1) of Γ at the vertex {1, 3} is isomorphic to the Clebsch graph, see also [13, pp.319], and 2. the vertex neighborhood subgraph Γ(2) of Γ(1) at the vertex {2, 3}, i.e., the induced subgraph of Γ on Γ1 ({1, 3})∩Γ1 ({2, 3}), is isomorphic to the triangular graph T (5), which is isomorphic to the complement of the Petersen graph. Graph (v, k, λ, µ) Petersen graph (10, 3, 0, 1) Clebsch graph (16, 5, 0, 2) Shrikhande graph (16, 6, 2, 2) Schläfli graph (27, 16, 10, 8) Gewirtz graph (56, 10, 0, 2) 1 T (n) ( 2 n(n − 1), 2(n − 2), n − 2, 4) L2 (n) (n2 , 2(n − 1), n − 2, 4) CP (n) (2n, 2(n − 1), 2(n − 2), 2) , q−5 , q−1 ) P (q) (q, q−1 2 4 4 For strongly regular graphs Γ with r = 0. Under this case, µ = k, the multiplicity k , see [12], and s = λ − k < 0. Let t = k − λ, k = lt for of r is a positive integer k−λ some positive integer l. Then s = −t, v = (l + 1)t, and (v, k, λ, µ) = ((l + 1)t, lt, (l − 1)t, lt). Straightforward combinatorial arguments show that Γ must be the complete multipartite graph Kt,t,...,t with (l + 1) equal parts of size t. Clearly any vertex neighborhood subgraph Γ(1) of Γ is again a strongly regular graph with parameters (lt, (l − 1)t, (l−2)t, (l−1)t). Furthermore, any vertex neighborhood subgraph Γ(i) of Γ(i−1) is still a strongly regular graph with parameters ((l−i+1)t, (l−1)t, (l−i−1)t, (l−i)t) for i = 1, 2, . . . , i−1, where Γ(0) = Γ. Each Γ(i) has distinct eigenvalues (l−i)t > 0 > −t. The complement of the ladder graph corresponds to the special case t = 2. Definition 11. 14.

(22) 1. The line graph L(Γ) of a graph Γ = (V, E) is defined on E ⊆ A, B ∈ E are adjacent if and only if |A ∩ B| = 1.. V 2. . , such that. 2. The line graph of the 2-(v 0 , k 0 , 1) design, Π = (X, B), is defined on B, such that A, B ∈ B are adjacent if and only if |A ∩ B| = 1. Definition 12. The Latin square graph is defined on n2 columns of the orthogonal array OA(k, n), which is a k × n2 array with entries from a set [n] such that the n2 ordered pairs defined by any two rows are all distinct, and two vertices adjacent if they have the same entries in one coordinate position. Graph Latin square graph on OA(k, n) Line graph. (v, k, λ, µ) (n , k(n − 1), n − 2 + (k − 1)(k − 2), k(k − 1)) 2. v(v−1) k(v−k) v−2k+1 ( k(k−1) , k−1 , k−1 + (k − 1)2 , k 2 ). of the 2-(v, k, 1) design Note that Petersen graph is the complement of L(K5 ), T (n) is L(Kn ) and L2 (n) is L(Kn,n ); the Latin square graph on OA(n − 1, n) has the same parameters as the complement of L2 (n). Theorem 2.4.2 tells us that some families of strongly regular graphs are uniquely determined by their parameters. Theorem 2.4.2. 1. Petersen graph is the unique SRG(10, 3, 0, 1); 2. Clebsch graph is the unique SRG(16, 5, 0, 2) [[19], pp.230]; 3. Schläfli is the unique SRG(27, 16, 10, 8); 4. T (n) is the unique SRG( n2 , 2(n − 2), n − 2, 4) if n > 8; 5. L2 (n) is the unique SRG(n2 , 2(n − 1), n − 2, 2) if n > 4. Proof. We only proof the uniqueness of Petersen graph. Let Γ be a strongly regular graph with parameters (10, 3, 0, 1). 1. Because every pair of adjacent vertices has 0 common neighbor, there is no triangle in Γ. 2. Because every pair of nonadjacent vertices has a unique common neighbor, Γ has no diamond as its induced subgraph. 15.

(23) Let {1, 2, 3, ..., 10} be the vertex set of Γ. Because Γ is 3-regular, there are three distinct vertices adjacent to vertex 1. Without loss of generality, let the neighborhood of vertex 1 be Γ1 (1) = {2, 3, 4}. Since there is no triangle in Γ, the vertices in Γ1 (1) are pairwise nonadjacent. Since Γ has no diamond as its subgraph, each pair of vertices in Γ1 (1) are nonadjacent and has no other common neighbor except 1. Therefore, each vertex in Γ1 (1) is adjacent to two other vertices of Γ \ (Γ1 (1) ∪ {1}). Without loss of generality, let Γ1 (2) = {5, 6}, Γ1 (3) = {7, 8}, and Γ1 (4) = {9, 10}. By reason 1, the vertices in the same set are nonadjacent. Each vertex in Γ1 (2) ∪ Γ1 (3) ∪ Γ1 (4) needs to be adjacent to two more vertices. If the neighbors of vertex 5 are in the same set, then we get a diamond, a contradiction. Hence, one of its neighbors must be in {7, 8}, and the other in {9, 10}. Without loss of generality, we add edges {5, 7} and {5, 9}. By reason 2, vertex 5 and 6 have no common neighbor except vertex 2. Hence joining {6, 8} and {6, 10} is the unique choice. Similarly, vertex 5 and 7 have no common neighbors. Therefore, joining {7, 10} and {8, 9} is the unique choice. The final graph Γ is a strongly regular graph with parameters (10, 3, 0, 1), which is isomorphic to Petersen graph. For n = 8, there are three other graphs with same parameters of T (8), SRG(28, 12, 6, 4), known as the Chang graphs. Theorem 2.4.3 ([11], pp.63). A strongly regular graph with least eigenvalue −2 is one of the following: 1. T (n) for n ≥ 5, L2 (n) for n ≥ 3 or CP (n) for n ≥ 2; 2. the Petersen graph, the Shirkhande graph or the three Chang graphs; 3. the complement of the Clebsch graph or of the Schl¨ afli graph.. 16.

(24) Chapter 3. The Friendship Property and Strongly Regular graphs In this chapter, graphs with the property that each pair of distinct vertices has the same number λ of common neighbors are introduced. The friendship theorem shows the graphs with λ = 1. And the k-regular graphs with λ > 1 are SRG(v, k, λ, λ). Symplectic graphs and the Cayley graphs associated with bent functions, which will appear in Section 4.1, are the examples.. 3.1. A review of friendship theorem. In a 1983 survey article [20] entitled ”The Friendship Theorem and the Lover Problem”, Hammersley assembled material appeared previously in separate contexts and diverse guises, such as the solubility of Diophantine quadratic matrix equations; the construction of block designs, the existence of finite geometries, etc. Moreover, in stressing the graph-theoretic aspects of the matter, he adopted a rather different line from traditional treatments. He began with a special case of the love problem, known as the friendship theorem. It is not known who first raised the following problem or who gave it its human touch. The earliest published paper [35] to Hammersley’s knowledge was Wilf (1971), who cites an earlier unpublished account by Graham Higman in 1968. However, Van Lint [11, pp.45] referred this result to Erdös, Rényi and Sós (1966)[18]. Suppose in a group of at least three people we have the situation that any pair of persons have precisely one common friend. Then there is always a person (the ”politician”) who is everybody’s friend. Before tackling the proof, let us rephrase the problem in graph-theoretic terms. We interpret the people as the set of vertices V with |V | = n ≥ 3, and join two vertices by an edge if the corresponding people are friends. We tacitly assume that friendship is always two-ways, that is, if u is a friend of v, then v is also a friend of u, and further that nobody is his or her own friend. Note that there are graphs with this property as in the figure, where u is 17.

(25) the politician; in fact we will show that these ”windmill graphs” are the only graphs with this property. Theorem 3.1.1 (The Friendship Theorem). If Γ is a graph in which any two distinct vertices have exactly one common neighbor, then Γ has a vertex joined to all others. Suppose that, in a finite community of n people, any two distinct individuals have exactly one mutual friend. Then n cannot be even, although any odd n ≥ 3 is possible. Further, the people can be labeled V1 , V2 , ..., Vn such that, whenever 2 ≤ 2r < n, V2r and V2r+1 are friends of each other and of V1 and of nobody else. Thus V1 , the Dale Carnegie (1953) of the community, is everybody’s friend. By hypothesis, friendship is symmetric but not reflexive: if Vi is a friend of Vj , then Vj is a friend of Vi , but Vi cannot be friend himself. Love, on the other hand, may or may not be reciprocated, and may be narcissistic. The love problem is the generalization of the friendship theorem when the asymmetric and possibly reflexive relationship of love replaces the symmetric and non-reflexive relationship of friendship. Whereas the friendship theorem is completely solved, the love problem is largely unsolved. It is starting that such a combinatorial - sounding result seems to have no short combinatorial proof. There do exist proofs avoiding eigenvalues (see Hammersley [20, 1983]), but they require complicated numerical arguments to eliminate regular graphs. Wilf (1971) used the feasible condition of strongly regular graphs to eliminate regular graphs and combine the condition that unique common neighbor forbids 4-cycles to prove that there is a vertex joined to all others. Craig Huneke has a short proof to exclude regular graphs by counting walks and using modular arithmetic; it is no longer than the proof of the Integrality Condition. The resulting graph consists of some number of triangles sharing a vertex. If Γ is a regular graph, then it is a strongly regular graph with µ = λ = 1. From the feasible condition about multiplicity of eigenvalue mentioned in Chapter 2, it is easy to see that triangle is the only choice. It should be clear that in the presence of a politician only the windmill graphs are possible. Several proofs of the friendship theorem exist, but the first proof, given 18.

(26) by Paul Erdös, Alfred Rényi and Vera Sós, is still the most accomplished. Proof. (by Paul Erdös,Alfred Rényi and Vera Sós) Suppose the assertion is false, and Γ is a counterexample, that is no vertex of Γ is adjacent to all other vertices. To derive a contradiction we proceed in two steps. The first part is combinatorics, and the second part is linear algebra. 1. We claim that Γ is a regular graph, that is, d(u) = d(v) for any u, v ∈ V . Note first that the condition of the theorem implies that there are no cycles of length 4 in Γ as in the figure. Let us call this the C4 -condition. We first prove that any two non-adjacent vertices u and v have equal degree d(u) = d(v). Suppose d(u) = k, where w1 , w2 , ..., wk are the neighbors of u. Exactly one of the wi , say w2 , is adjacent to v, and w2 adjacent to exactly one of the other wi ’s, say w1 , so that we have the situation of the figure to the left. The vertex v has with wi the common neighbor w2 , and with wi (i ≥ 2) a common neighbor zi (i ≥ 2). By the C4 -condition, all these zi must be distinct. We conclude d(v) ≥ k = d(u), and thus d(u) = d(v) by symmetry. It remains to show that d(u) = d(v) holds also for adjacent vertices u and v. Let v = w1 , w2 , ..., wk be the neighbors of u. If any of the neighbors z of u (and there must be at least one by our assumption) is also non-adjacent to v, then we infer d(u) = d(z) = d(v) by what we just proved. Hence we may assume that v is adjacent to all z ∈ / {w2 , ..., wk }. By our assumption d(v) < n − 1 there must be some wi , say w2 , which is not adjacent to v. But z1 and w2 must have a common neighbor. It cannot be u since u and z1 are not adjacent, nor can it be v = w1 since v and w2 are not adjacent. It cannot be any of the other wj by the C4 -condition, and this has exhausted all the possibilities of a common neighbor of z1 and w2 . In conclusion, d(u) = k for all u, for some k between 2 and n − 2. Looking at the figure for the case of adjacent vertices again, we find n = k 2 − k + 1. Indeed, any of the wi ’s have exactly k − 2 neighbors outside {w1 , w2 , ..., wk } has a common neighbor with u. Hence n = 1 + k + k(k − 2) = k 2 − k + 1.. (1). 2. The rest of the proof is a beautiful application of some standard results of linear algebra. Note first that k must be greater than 2, since k = 2, only Γ = K3 is possible by (1), which is a windmill graph. Consider the adjacency matrix A = (aij ). By part 1, any row has exactly k 1’s, and by the condition 19.

(27) of the theorem, for any two rows there is exactly one column where they both have a 1. Note further that the main diagonal consists of 0’s. Hence we have   k 1 ... 1    1 k ... 1  2 A = .. . . ..   ..  = (k − 1)I + J . . . .   1 ... 1 k where I is the identity matrix, and J the matrix of all 1’s. It is immediately checked that J has the eigenvalues n (of multiplicity 1) and 0 (of multiplicity n − 1). It follows that A2 has the eigenvalues k − 1 + n = k 2 (of multiplicity 1) and k − 1 (of multiplicity n − 1). Since A is symmetric and hence diagonalizable, we conclude that A has the √ eigenvalue k (of multiplicity 1) and ± k − 1. Suppose r of the eigenvalues are √ √ equal to k − 1 and s of them are equal to − k − 1, with r + s = n − 1. Now we are at most home. Since the sum of the eigenvalues of A equals the trace √ √ (which is 0), we find k + r k − 1 − s k − 1 = 0, and, in particular, r 6= s, √ √ √ k and k − 1 = s−r . It follows that k − 1 is an integer h (if m is rational, then it is an integer!), and we obtain h(s − r) = k = h2 + 1. Since h divides h2 + 1 and h2 , we find that h must be equal to 1, and thus k = 2, which we have already excluded. So we have arrived at a contradiction, and the proof is complete.. A proof was given by D. G. Higman [22]. Another quite elementary proof, though not wholly elementary, was given by Wilf. Wilf assumed the conclusion is false and defined an incidence structure which is a projective plane. Finally, he used the Integrality Condition, which West used to eliminate the regularity of the graph, to get the nonexistence of the structure. Proof. ([35], Wilf 1971) We assume that the conclusion is false. Then for our graph we have the hypotheses 1. Each pair of different x, y has a unique common neighbor F (x, y). 2. Each point x, there exists a y different from x and y not adjacent to x. 20.

(28) We then define an incidence structure Π = (X, L) associated with the graph under consideration with X the vertex set V (Γ) of the graph Γ, and L = {l(x) : x ∈ X} where l(x) = {y : y ∈ X is adjacent with x}. We shall claim that Π = (X, L) is a finite projective plane, indeed, it suffices to show that there is a set of four points of Π, no three of which lie on a line. We may assume that n ≥ 4 by hypotheses 1 and 2. By 1, every pair of points lies on exactly one line, indeed {x, y} ∈ l(F (x, y)), and every pair of lines of Π have exactly one point in common, namely l(x) ∩ l(y) = {F (x, y)}. Choose four distinct points of Π. If no three lie on a line we are finished. Otherwise, some three have a common neighbor a. By 2, there is a b such that a and b are not adjacent. If F (a, b) is the common neighbor of a and b, let z be any neighbor of a other than F (a, b) and F (F (a, b), a). Then we have the figure 3.1. In the picture, a solid line denotes ”adjacency” and a dotted line ”non-adjacency”. We claim that four distinct points F (a, b), a, z, b satisfy the condition that no three of them lie on a line. 1. If F (a, b), a, z have a common neighbor, it is F (a, F (a, b)). But z and F (a, F (a, b)) are not adjacent for otherwise z and F (a, b) would have two common neighbors. 2. If F (a, b), a, b have a common neighbor, it is F (a, b), a contradiction. 3. If F (a, b), z, b have a common neighbor, since a = F (z, F (a, b)), and a, b are not adjacent, we have a contradiction. 4. If a, z, b have a common neighbor, it is F (a, b). But z and F (a, b) are not adjacent because otherwise F (a, b) and a have two common neighbors, namely z and F (a, F (a, b)). It follows that the structure Π is a finite projective plane of order m for some positive integer m. Hence Π has m2 + m + 1 points and m2 + m + 1 lines, that each line of Π contains m + 1 points and that each point of Π is on m + 1 lines. Let A denote the incidence matrix of the points and lines of Π. Then A has the properties 21.

(29) that A is symmetric with trace(A) = 0, A2 is the matrix with entry m + 1 in all diagonal positions and 1 in all off-diagonal positions, a contradiction, since that, there exists no projective plane of order m > 1 whose incidence matrix is symmetric 2 with trace zero, is proved as follow. The spectrum of A2 is ((m2 + m + 1)1 , mm +m ), √ µ √ µ and hence the spectrum of A is (m + 11 , m 1 , − m 2 ). Clearly, µ1 + µ2 = m2 + m, √ √ √ trace(A) = (m + 1) + µ1 m − µ2 m = 0, and then µ2 = 12 (m2 + m + m + √1m ) is not an integer. Proof. [34, pp.467] The symmetry of the condition suggests that Γ might be regular. If Γ is a regular graph, then it is strongly regular with λ = µ = 1. By the Integrality Condition, k k ) is an integer. Hence √k−1 is an integer, which this requires that 21 ((n − 1) ± √k−1 happens only when k = 2. However, K3 is the only 2-regular graph satisfying the condition, and it does have vertices of degree n − 1. Now suppose Γ is not regular. We show that any two non-adjacent vertices have the same degree. Insistence on unique common neighbors forbids 4-cycles. Suppose v and w are not adjacent, and let u be their common neighbor. Let a be the common neighbor of u, v and b the common neighbor of u, w. We want to show w has as many neighbors as v. Any x ∈ S = Γ1 (v) − u − a has a common neighbor f (x) with w. If f (x) = b for any x ∈ S, then x, b, u, v is a 4-cycle. If f (x) = f (x0 ) for distinct x, x0 ∈ S, then x, v, x0 , f (x) is a 4-cycle. Hence w has distinct neighbors for each neighbor of v, and d(w) ≥ d(v). By symmetry, d(v) ≥ d(w). Since Γ is not regular, it has two vertices v, w with d(v) 6= d(w). By the preceding argument, we know v ↔ w. Let u be their common neighbor. Since u cannot have the same degrees each of them, we may assume d(u) 6= d(v). Now suppose Γ has a vertex x not adjacent to v. Then d(x) = d(v), but this requires x ↔ w and x ↔ u. This creates the 4-cycle v, u, x, w. Hence d(v) = n − 1. Let us rephrase our theorem in the following way: Suppose Γ is a graph with the property that between any two vertices there is exactly one path of length 2. Clearly, this is an equivalent formulation of the friendship condition. Our theorem then says that the only such graphs are the windmill graphs. But what if we consider paths of length more than 2? A conjecture of Anton Kotzig asserts that the analogous situation is impossible. Kotzig’s conjecture has been verified for some l, but the general case remains open. Kotzig’s Conjecture. Let l ≥ 2. Then there are no graphs with the property that between any two vertices there is precisely one path of length l.. 22.

(30) 3.2. SRG with µ = λ. We now consider SRG(v, k, λ, λ). When λ = 1, K3 is the only choice for regular graphs which has been proved in friendship theorem. And there are finitely many graphs with λ > 1 which can be known in Theorem 3.2.1. The symplectic graphs Sp(2m) in Section 3.3 offer a family of such strongly regular graphs with parameters (22m − 1, 22m−1 , 22m−2 , 22m−2 ) for positive integers m, note that K3 is the symplectic graph Sp(2). The Cayley graphs associated with bent functions in Section 4.1 provide another family of such graphs. Some necessary conditions among v, k, λ and their spectrums are given in the following theorem. Theorem 3.2.1. Suppose there exists a SRG(v, k, λ, λ) with λ > 1, and with Spec(Γ) = (k 1 , θmθ , τ mτ ), then √ 1. θ = −τ = k − λ, θτ = −(k − λ) are integers with multiplicities mθ = 1 k k ), and mτ = 12 ((v − 1) + √k−λ ) respectively. ((v − 1) − √k−λ 2 2. θ | λ and (v, k) = ( (θ. 2 +θ+λ)(θ 2 −θ+λ). λ. , θ2 + λ).. 3. for each λ, there are only finitely many feasible parameter sets. 4. if θ = λ then (v, k, λ) = (λ2 (λ + 2), λ(λ + 1), λ). k , which is a positive integer by 1. Hence Proof. 1. Omitted. 2. Let t = √k−λ √ √ 2 2 k = t ±t 2t −4λ , both t and b = t2 − 4λ are of the same parity; since t2 − 4λ = b2 , it follows that 4λ = (t + b)(t − b), q k k t + b = √k−λ + ( √k−λ )2 − 4λ and q k k √ )2 − 4λ t − b = k−λ − ( √k−λ. must be even. Let t + b = 2h1 and t − b = 2h2 for some positive integers h1 > h2 , hence λ = h1 h2 , then t = h1 + h2 , b = h1 − h2 , and k is either h1 (h1 + h2 ) or √ h2 (h1 + h2 ). Note that θ = k − λ is either h1 ( in case k = h1 (h1 + h2 )) or h2 (in 2 2 −θ+λ) in either case case k = h2 (h1 + h2 )), hence θ | λ. It follows that v = (θ +θ+λ)(θ λ as required. √ Since θ = −τ as shown in Theorem 3.2.1, θ2 = k − λ < k − 1 and θ < 2 k − 1, a SRG(v, k, λ, λ) turns out to be a Ramanujan graph, see details in Section 4.2. Indeed, the above theorem paves a way for studying possible feasible parameters (v, k, λ, λ) for a given λ with a pair (h1 , h2 ) either (θ, λθ ) or ( λθ , θ). The trivial decomposition of λ = 1 · λ with (h1 , h2 ) = (λ, 1) leads to (v, k, λ) = (λ2 (λ + 2), λ(λ + 1), λ) or (λ + 2, λ + 1, λ). 23.

(31) √ Another extremal cases with h1 , h2 closed to λ are considered for λ = 22m and 2m (2m + 1) respectively. If λ = 22m with (h1 , h2 ) = (2m , 2m ), then (v, k, λ) = (22m+2 − 1, 22m+1 , 22m ) which is identical with those of the symplectic graphs; if λ = 2m (2m + 1) with (h1 , h2 ) = (2m + 1, 2m ), then (v, k, λ) = (22 (2m + 1)2 , (2m + 1)(2m+1 + 1), 2m (2m + 1)) or (2m (2m+2 ), 2m (2m+1 + 1), 2m (2m + 1)); and the former type is realized by a set of 2m M OLS of order 2m+1 + 2, called Latin square graphs. Graphs Symplectic graphs Sp(2m + 1) Cayley graphs associated with bent functions over Z22m+2 Latin square graphs 2. Parameters (22m+2 − 1, 22m+1 , 22m ) (2m (2m+2 ), 2m (2m+1 + 1), 2m (2m + 1)) (22 (2m + 1)2 , (2m + 1)(2m+1 + 1), 2m (2m + 1)). 2. −θ+λ) is an integer, λθ is a divisor of θ(θ2 − 1). In the previous Since v = (θ +θ+λ)(θ λ theorem, there are too many methods to factor λ as the product of two positive 2 2 −θ+λ) , θ2 + integers if λ is not a prime. For each (θ, λ) with θ|λ, (v, k, λ) = ( (θ +θ+λ)(θ λ λ, λ) needs not to be a feasible parameter set for SRG(v, k, λ, λ); for example (θ, λ) = 2 2 −θ+λ) = 27·21 is not an integer. However, if θ = 2 is an eigenvalue (3, 15). (θ +θ+λ)(θ λ 15 of a SRG, and λ = 2 · p for some prime p, then v = 16 is the unique choice and (v, k, λ) = (16, 6, 2). mτ. Corollary 3.2.2. If Γ is a SRG(v, k, λ, λ) with Spec(Γ) = (k 1 , θmθ ,τ ) and λ = θ·q, √ then q|θ(θ2 − 1). Moreover, θ = cq + 1 for some integer c if gcd(θ, q) = 1, and in general dq |θ2 − 1 where d = gcd(θ, q). Proof. It is easy to be proved by checking the integrality condition of v. Theorem 3.2.3. Suppose there exists a SRG(v, k, λ, λ) with Spec(Γ) = (k 1 , θmθ , τ mτ ) and λ = p · q for distinct primes p > q. , p) , and p = 2cq ± 1 for some integer c. 1. If q ≥ 3, then (v, θ) = ( p(p+q−1)(p+q+1) q , p). 2. If q = 2, then (v, θ) = (16, 2) or ( p(p+1)(p+3) 2. 24.

(32) by Theorem 3.2.1. Since p, q are primes and v is an Proof. Let v = q(p+q−1)(p+q+1) p integer, (p + q − 1)(p + q + 1) ≡ 0(mod p), and hence q 2 ≡ 1(mod p), and hence q ≡ 1 or − 1(mod p). Because p is a prime, it follows that q = cp ± 1 for some even integer c. If 3 ≤ q < p, then q = 1 or p − 1, a contradiction. Because p and q are odd . It is easy to primes and p = cq ± 1 for some even integer c if v = p(p+q−1)(p+q+1) q check that θ = p by theorem 4.1. For q = 2, since p is odd, (p + 1)(p + 3) is even, then either (v, θ) = ( p(p+1)(p+3) , p) 2 2(p+1)(p+3) , 2). The only choice for p in the later case is 3, and hence or (v, θ) = ( p (v, θ) = (16, 2). In Theorem 3.2.2, we know that a SRG(v, k, λ, λ) with τ = −2, then (v, k, λ) = (16, 6, 2). In general case for any λ and µ, we have a theorem which decides all the possible strongly regular graphs with at least eigenvalue −2 in Section 2.4, see details in [11].. 3.3. Symplectic graphs. Let N be a block diagonal matrix with m blocks of the form. 0 1 1 0. ! .. Definition 13. The symplectic graph has vertex set V (Sp(2m)) = GF (2)2m − {0}, and u and v are adjacent if and only if uT N v = 1. The binary rank of Sp(2m) is 2m from the order of N . The name ”symplectic graph” arises because the function f (u, v) = uT N v is known as a symplectic form. Two vertices u and v are orthogonal with respect to f if f (u, v) = 0. Therefore, Sp(2m) is the non-orthogonality graph of GF (2)2m − {0} with respect to the symplectic form f . Actually every graph ,which has no isolated vertices with the property that the neighborhoods of distinct vertices are distinct, has the adjacency defined as above by relabelling the vertices, and we can compute their binary ranks. Here are some properties between Sp(2m) and its induced subgraphs. Theorem 3.3.1 ([19], pp.184-185). 1. Suppose Γ has no isolated vertices and the neighborhoods of distinct vertices are distinct. Then Γ has binary rank at most 2m if and only if it is an induced subgraph of Sp(2m). 2. Every graph on 2m − 1 vertices occurs as an induced subgraph of Sp(2m).. 25.

(33) This implies that studying the properties of the universal graph Sp(2m) can yield information that applies to all graphs with binary rank 2m. A trivial observation of this kind is that a reduced graph with binary rank 2m has at most 2m − 1 vertices. Since the chromatic number of an induced subgraph is less then the chromatic number of the origin graph, we have a theorem as follow. Theorem 3.3.2 ([19], pp.243-244). 1. The chromatic number of Sp(2m) is 2m + 1. 2. For any graph Γ with binary rank 2m, χ(Γ) ≤ 2m + 1. Finally, the Sp(2m) is a strongly regular graph with µ = λ, and the parameter set, which can be proved from the theory of linear algebra, and its spectrum are as follow theorem. Theorem 3.3.3 ([19], pp.243). The graph Sp(2m) is SR(22m − 1, 22m−1 , 22m−2 , 2m−1 −2m−1 −1 2m−1 +2m−1 −1 22m−2 ) and with Spec(Sp(2m)) = ((22m−1 )1 , (2m−1 )2 , (−2m−1 )2 ). Proof. Because the rank of y T N is 1, applying rank(y T N ) + nullity(y T N ) = 2m to get nullity(y T N ) = 2m − 1; the number of nonzero vector x satisfying y T N x = 1 is 22m − 22m−1 = 22m−1 . The other two parameters of SRG are both 22m−2 .. 26.

(34) Chapter 4. Bent Functions, Ramanujan graphs and SRG Motivated from Ramanujan graphs and the spectra of Cayley graphs associated with bent functions, the Cayley graphs associated with bent functions are all SRG with µ = λ, we then show that all SRG with µ = λ are Ramanujan graphs, providing a practical model for communication.. 4.1. SRG associated with bent functions. From design theory, there are many methods used to associated a 2-design with a graph, like block graph, line graph, and viewing the incidence matrix of a symmetric 2-design as an adjacency matrix of a graph. In group theory, there is also a method used to construct a graph, called Cayley graph, from a finite group. Definition 14. Given a finite group G and a subset C of nonidentity elements of G such that α ∈ C implies that α−1 ∈ C, the Cayley graph is the simple graph with vertex set G and where vertices α and β are adjacent if and only if βα−1 ∈ C. A complete graph Kn is a Cayley graph with respect to any group of order n, where C consists of all nonidentity elements. Often it is required that C generates the group; this ensures that the corresponding Cayley graph is connected. We will introduce a Cayley graph associated to a Boolean function f : Zn2 → Z2 . Whose vertices, elements of Zn2 , are assigned the elements of Z2 by f and the Cayley set C is chosen as the collection of x ∈ Zn2 which is assigned 1. The problem of analyzing the spectral coefficients of Boolean functions has been brought to the framework of spectral analysis of graphs though their associated Cayley graphs, and hence the using of tools from algebraic graph theory for investigations related to the spectral coefficients of Boolean functions with small numbers of distinct coefficients is possible. Among others, a characterization of bent functions in terms of strongly regular graphs by Bernasconi, Codenotti, and VanderKam [5, 6] is a successful example. It was shown in [5] that the associated Cayley graph of a bent function is a strongly regular graph by showing that it has exactly three 27.

(35) distinct eigenvalues. They further showed that bent functions are the only Boolean functions f with associated strongly regular graph by studying the integral solutions of a quadratic equation in [6]. As a consequence, bent functions can be characterized as Boolean functions with a certain class of strongly regular graphs, followed by a nice interpretation of bent functions in terms of strongly regular graphs. The Fourier transform of a Boolean function f (x) : Zn2 → Z2 is defined to be f ∗ (w) =. 1 X f (x) · (−1)hw,xi 2n ∀x∈Zn 2. which satisfies the property that f (x) =. 1 X ∗ f (w) · (−1)hw,xi . n 2 ∀w∈Zn 2. The Cayley graph Γf associated with a Boolean function f : Zn2 → Z2 is defined on the vertex set Zn2 , with u, w ∈ Zn2 adjacent if w ⊕ u ∈ Ωf = f −1 (1) or equivalently f (w ⊕ u) = 1. For a Boolean function f : Zn2 → Z2 , the spectrum of Γf is usually denoted by Spec(Γf ) = (|Ωf |, θ1 , ..., θ2n −1 ) where θi =. X. f (x) · (−1)hb(i),xi = 2n · f ∗ (b(i)). ∀x∈Zn 2. and b(i) is the binary representation of i; the multiplicity of its largest eigenvalue f ∗ (b(0)) is 2n−dimhΩf i (which implies the graph Γf is |Ωf |-regular with 2n−dimhΩf i connected components and the graph Γf is connected if dimhΩf i = n). A Boolean functions is characterized by its spectrum if it is possible to identify its associated graph (i.e., determine all the details of its topology) only on the basis of the knowledge of its distinct eigenvalues, i.e., without using any information regarding their eigenvectors. It is interesting to note that the fewer the number of distinct spectral coefficients are, the stronger are the algebraic properties of the set Ωf ; for instance, it is well-known that if a connected graph has exactly m distinct eigenvalues, then its diameter d satisfies d ≤ m − 1. A Boolean function f : Zn2 → Z2 is called a bent function if 1 ((−1)f (x) )∗ (w) = ± √ 2n for any w ∈ Zn2 . It is equivalent to say that f is a bent function if n. n. 2n · f ∗ (0) = 2n−1 − 2 2 −1 or 2n−1 + 2 2 −1 and n 2n · |f ∗ (w)| = 2 2 −1 for any w 6= 0, 28.

(36) the term of bent was coined by Rothaus [30]. If f (x) is a bent function on Zn2 , n ≥ 3, then n = 2k must be even, and the degree of f (x) is at most k; moreover f (x) is irreducible whenever deg(f (x)) = k ≥ 3, see [30] for details. The existence of bent functions f (x) is equivalent to the fact that whether [(−1)f (x+y) ] is a Hadamard matrix. A k-regular connected graph is strongly regular if and only if it has exactly three distinct eigenvalues θ0 = k, θ, τ , with multiplicities 1, mθ , and mτ respectively. Pn A rephrase of Parseval’s identity gives that f ∗ (b(0)) = 2i=0−1 (f ∗ (b(i)))2 and then yields the following useful equality (k − θ)(k − τ ) = 2n (k + θτ ) where k = |Ωf |, and n must be replaced by dimhΩf i if Γf is not connected. If Γf is a SRG(v, k, λ, µ), then λ = k + θτ + θ + τ and µ = k + θτ . It was also observed that the class of bent functions is associated to a very special class of strongly regular graphs, and indeed identifies the bent functions precisely. If f is a Boolean function on Zn2 with connected strongly regular graph Γf , then there exists y ∈ Ωf such that x ⊕ y ∈ Ωf for each x ∈ Zn2 \ Ωf , and there exist h elements z ∈ Ωf such that y ⊕ z ∈ Ωf , where h = λ if y ∈ Ωf , and µ if y ∈ / Ωf for n each y ∈ Z2 . In order to find a complete characterization of the class of functions with three distinct nonzero spectral coefficients with additional properties, it was proved in [6] that the quadratic equation x2 − 2n x + (2n − 1)y 2 = 0 has integer solutions in x and y only if y 2 = 0, 1, 2n−2 . As a consequence, bent functions can be characterized as binary functions with a certain class of strongly regular graphs. Theorem 4.1.1 ([5, 6]). The associated Caley graph Γf of a bent function f is a SRG(v, k, λ, λ); moreover, the bent functions are the only Boolean functions f whose associated graph Γf is a SRG(v, k, λ, λ) Those graphs Γf with small numbers of distinct eigenvalues are considered: if Γf has a single eigenvalue, then Γf = K2n −1 ; if Γf has two distinct eigenvalues, then n n Γf is either |Ωf2 |+1 K|Ωf |+1 when b(0) ∈ / Ωf , or |Ω2 f | K|Ωf | with loops otherwise; if Γf has three eigenvalues, then (k, θ, τ ) = (|Ωf |, 0, −|Ωf |) if and only if Γf is the complete bipartite graph between vertices in Ωf and in Zn2 \ Ωf ; (k, θ, τ ) = (|Ωf |, 0, τ ) if |Ω | and only if Γf is a complete multipartite graph with Γf = (− τf + 1)K−τ . If Γf is connected, then Γf is a SRG(2n , |Ωf |, λ, µ) with −τ (2n −1)−|Ωf | −θ(2n −1)+|Ωf | √ √ 1 ) 1 ) θ−τ θ−τ , ( (λ − µ − ∆))( ) Spec(Γf ) = (|Ωf |1 , ( (λ − µ + ∆))( 2 2 where ∆ = (λ − µ)2 − 4(µ − |Ωf |).. Theorem 4.1.2. If f is a bent function with connected Γf , then Γf is a SRG(v, k,. 29.

(37) λ, λ) with (v, k, λ) is either n. n. n. n. (2n , 2n−1 + 2 2 −1 , 2n−2 + 2 2 −1 ) or (2n , 2n−1 − 2 2 −1 , 2n−2 − 2 2 −1 ) and with spectrum Spec(Γf ) either n. n. n −1 2. n −1 2. ((2n−1 + 2 2 −1 )(1) , (2 2 −1 )(2 or. ((2n−1 − 2. )(1) , (2. n n−1 −2 2 −1 −1). n (2n−1 −2 2 −1 ). ). n. n n−1 +2 2 −1 ). n −1 2. n (2n−1 +2 2 −1 −1). , (−2 2 −1 )(2. , (−2. ). ) ). respectively. It has been shown that certain Boolean functions, depending on say, k variables, may be classified as equivalent under the set of affine transformations. The number of equivalent classes is much smaller than the total number of functions depending on k variables. For instance, the total number of Boolean functions depending on 4 variables is 65,536, while the number of canonic functions on four variables under affine transformations is only eight, while only the 8th one represents a bent function. 1. The first canonic function is simply the constant function. Hence its associated graph has only one eigenvalue 0, and is totally disconnected graph. 2. The second canonic function coincides with the AND function. The spectrum of its associated graph is (18 , −18 ), which has eight connected components, is simply a matching. 3. The graph associated to the third canonic function has spectrum (24 , 08 , −24 ), it follows that each connected components is a complete bipartite graph, i.e., 4K2,2 . 4. The graph associated the fourth canonic function has spectrum (32 , 16 , −16 , −32 ), it follows that it has two connected components and each corresponding to a three dimensional cube. 5. The graph associated the fifth canonic function has spectrum (42 , 012 , −42 ), it follows that it has two connected components and each component is a complete bipartite graph, i.e., 2K4,4 . 6. The graph associated the sixth canonic function has spectrum (41 , 24 , 06 , −24 , −41 ) and is a connected bipartite graph with valency 4 and diameter 4. 7. The graph associated the seventh canonic function has spectrum (51 , 110 , −35 ), which is the Clebsch graph, the unique SRG(16, 5, 0, 2).(See figure ??) 30.

(38) Figure 4.1: L2 (4) 8. The graph associated the eight canonic function is a bent function and has spectrum (61 , 26 , −29 ). It follows that the associated graph is a SRG(16, 6, 2, 2) which is isomorphic to L2 (4).. 4.2. SRG which are Ramanujan graphs. Telecommunications networks are frequently modeled using graph theory. Such networks have to transmit information quickly, so the question of how long it takes news to spread through the network is a central one. News spreads rapidly through the network if, for each subset X of the vertex set, there are many neighbors of the vertices of X that are not themselves in X. Let Γ be a k-regular graph of order n, and consider the eigenvalue of its adjacency matrix. The largest eigenvalue of Γ is k; let θ(Γ) be the absolute value of the next to largest eigenvalue. An asymptotic √ lower bound for θ, due to Alon and Boppana, states that lim inf θ(Γ) ≥ 2 k − 1. n→∞ This lower bound gives rise to the following definition. √ Definition 15. A k-regular graph Γ is a Ramanujan graph if θ(Γ) ≤ 2 k − 1. Theorem 4.2.1 ([15]). The diameter d(Γ) of a k-regular graph Γ of order n satisfies log(n − 1) the inequality d(Γ) ≤ d e. log( kθ ) Since, for fix k, θ is asymptotically minimal for Ramanujan graphs, these graphs also minimize the diameter. Since the diameter of a graph corresponds to the worstcase transmission time in the corresponding telecommunications networks, the Ramanujan graphs provide models of communications networks that are extremely good for both average and worst-case transmission times. We will see in the following theorem that Moore graphs and strongly regular graphs with µ = λ are Ramanujan graphs. Theorem 4.2.2. Suppose Γ is a SRG(v, k, λ, µ) with Spec(Γ) = (k 1 , θmθ , τ mτ ), 3k − 4 . Then Γ is a Ramanujan graph. k ≥ 2 and |λ − µ| ≤ √ 2 k−1 31.

(39) p Proof. If λ > µ, which implies θ > |τ |, implies (λ − µ) + (λ − µ)2 + 4(k − µ) < p √ 4 k − 1 or λ < µ, which implies θ < |τ |, implies (λ − µ)2 + 4(k − µ) − (λ − µ) < √ 4 k − 1, then we can say that Γ is also a Ramanujan graph. p √ Let λ − µ = t ≥ 0. Suppose a contradictory, t2 + 4(k − µ) > 4 k − 1 − t. √ √ √ t2 +4(k−µ) > 16(k−1)+t2 −8t k − 1, since 0 ≤ t ≤ 23k−4 implies 4 k − 1−t > 0. k−1 √ √ , a contradiction. It is similar as Hence 0 < µ < −3k + 4 + 2t k − 1 and t > 23k−4 k−1 t < 0.. 32.