Theorem 4.4.1 ([7], Theorem 2) Suppose that Γ is a strongly regular multigraph with parameters (m, n, µ, γ, R) with geometric parameter (r, k, t, c), i.e., (m, n, µ, γ, R) = this goal, we will find an upper bound forP
x∈V sxwhere sx :=P
x∈C
P
C6=C0|C ∩C0|2.
We then derive Lemmas 4.4.2 ∼ 4.4.7 in terms of elementary counting techniques and taking advantage of some inequalities.
Find an upper bound for Pµx
i=1(|Ci| − 1), and in particular its exact value whenever µx= r (Lemma 4.4.2), and upper bound for the size of maximal cliques (Lemma 4.4.4) and their lower bound when they contain a vertex x with µx = r (Lemma 4.4.3). We then find an upper bound for P
C6=C0∈M|C ∩ C0|2 in terms ofP
x∈V(µx− r) (Lemma 4.4.5). Moreover, an upper bound for P
x∈V(sx− 3rtk(µx− r)) (Lemma 4.4.6), lower bounds for µxsxin terms of Pµx
i=1(|Ci| − 1)2 and max{|Ci||i 6 µx} (Lemma 4.4.7 (1,2)) respectively; also lower bound for sx (Lemma 4.4.7 (3)) and for µxsx (Lemma 4.4.7 (4)) for vertex x with µx= r, and µx > r respectively.
We show that there exist at most 2rk vertices x satisfying µx > r (i.e., |Vs| = vs 6 2rk, Lemma 4.4.8) by studying upper and lower bounds of tx − s0 where tx := sx − 3rtk(µx − r) and s0 := (r − 1)tk − (r − 1)t2. We then show that each
point x with µx > r lies in at least five normal cliques consisting of all points y with µy > r (Lemma 4.4.9) by a contradictory argument together with some inequalities.
Moreover, we show |C1∩ C2| 6 µ for any distinct maximal cliques C1, C2; and then show that C − (S
C0∈NC0) is nonempty for any subfamily N of maximal cliques with
|N| 6 4r and C ∈/ N (Lemma 4.4.10).
We first claim that µx = r is equivalent to the emptiness of the set Ms:= {C|∀x ∈ C, µx > r} by Theorem 4.3.1 and Lemma 4.4.9, we then claim that Ms is empty in terms of the principle of inclusion and exclusion and some inequalities by contradic-tory argument over he conditions |N| 6 4r and the fact that each point x with µx> r lies in at least five cliques.
Suppose that there exists an integer a such that Theorem 4.3.1 (a,b,c) and |C| >
k − aµ for each C ∈ M, and m 6 a 6 2m − 1 are fulfilled.
Let m = r, n = k + r + c − 1 − t, µ = rt, γ = rc, R = r(k − 1). Assume that k > rt(4a + r + 5), k > (c + 1)t, r > 3 and r(c + r − 1) 6 (r − 1)t.
The relation of Lemma 4.4.2 ∼ Lemma 4.4.11:
Lemma 4.4.5 Lemma 4.4.6
Lemma 4.4.4 Lemma 4.4.7 (2) Lemma 4.4.7 (3) Lemma 4.4.8 Lemma 4.4.7 (1) Lemma 4.4.2 Lemma 4.4.7 (4)
Lemma 4.4.3 Lemma 4.4.9 Lemma 4.4.10
Lemma 4.4.11
Figure 3: Section 3 in [7], Metsch
Lemma 4.4.2 ([7], Lemma 3.10) l := µx, x ∈ C1, C2, . . ., Cl. Then (1) Pl
i=1(|Ci| − 1) 6 r(k − 1) +12γ;
(2) Pl
i=1(|Ci| − 1) = r(k − 1) if l = r.
From Pl
i=1(|Ci| − 1) = r(k − 1), we have:
Lemma 4.4.3 ([7], Lemma 3.12) If µx = r and x ∈ C, then |C| > k − (r − 1)(c + 1).
Lemma 4.4.4 ([7], Lemma 3.11) ∀C ∈ M, |C| < k + c + 1.
Lemma 4.4.5 ([7], Lemma 3.13)P
C6=C0∈M|C ∩C0|2 6 v(r −1)t+3r(c+1)P
x∈V(µx− r).
Lemma 4.4.6 ([7], Lemma 3.14)P
x∈V(sx− 3rtk(µx− r)) 6 vt(r − 1)(k + c + 1).
Lemma 4.4.7 Let l := µx, x ∈ C1, C2, . . . Cl. Let zi := |Ci| − 1, i = 1, 2, . . . , l, and z := max{zi|i = 1, 2, . . . , l}. Then
(1) l · sx > (n − m)(R + γ) + Rµ − 2(1 + l − m)zγ −Pl
i=1zi2 ([7], Lemma 3.15);
(2) l · sx> r(k − 1)2+ r(r − 1)tk − r(r − 1)t − rtc − r(c + 1)2− 2γ(k + c)(l − r) −Pl
i=1z2i ([7], Lemma 3.16);
(3) sx > (r − 1)tk − (r − 1)t2 for every vertex x satisfying µx = r ([7], Lemma 3.17);
and
(4) µxsx > r(r − 1)tk + k(k − 4µ − aµ)(µx− r) for every vertex x satisfying µx > r ([7], Lemma 3.18).
Lemma 4.4.8 ([7], Lemma 3.19) There exist at most 2rk vertices x satisfying µx > r, i.e., vs6 2rk.
Lemma 4.4.9 ([7], Lemma 3.20) Every vertex x satisfying µx > r lies in at least five cliques, which contain only vertices y satisfying µy > r.
Lemma 4.4.10 ([7], Lemma 3.21) (1) |C1∩ C2| 6 µ for C1 6= C2.
(2) C ∈ M, N ⊆ M with |N| 6 4r and C ∈/ N , then there exist x ∈ C and x ∈/
C0 ∈ N for all C0.
Lemma 4.4.11 ([7], Lemma 3.22) µx = r for each vertex x.
5 A Class of Strongly Regular Multigraphs
In this section, we will use the special properties of the definition of alternating form graph to define a symmetric association scheme. From the symmetric association scheme, we have that the alternating form graph is a distance regular graph. At last, we will define a graph Γ which is the induced subgraph of the alternating form graph, and give the multiplicity on the edges, and we will get a class of strongly regular multigraphs.
Definition 5.1 The alternating form graph Alt(n, q) is the simple graph with ver-tex set V = {A|A ∈ Mn×n(GF (q)), A = −AT} and the edge set E = {(A, B)|A, B ∈ V, rank(A − B) = 2}.
Since the alternating form graph Alt(n, q) is defined on the set of all skew-symmetric n × n matrices over GF (q), the rank of A − B is 2i for any two matrices A and B in Alt(n, q). Let Ri = {(A, B)|A, B ∈ Alt(n, q), rank(A − B) = 2i}, the relation classes
{R0, R1, . . . , Rd}, defined on Alt(n, q) × Alt(n, q), where d = dn2e; then (1) (Alt(n, q), (Ri)di=0) is a symmetric association scheme.
(2) (Alt(n, q), R1) is a distance regular graph with the intersection numbers {b0, b1, . . . , bd−1; c0, c1, . . . , cd}.
Theorem 5.2 Each maximal cliques of the alternating form graph Alt(n, q) is either
isomorphic to
respec-tively, called typed I and type II respectively.
For studying possible combinatorial geometric structures over Alt(n, q) , we shall study the matrix representations of those maximal cliques of both types containing the zero form. The others may be obtained simply by translation. Without loss of generality, we may assume that V is an inner product space with a fixed orthonormal basis {v1, v2, . . . , vn}. If v ∈ V is a nonzero vector, then < v >=< αv > for all
where αj ∈ GF (q), 1 6 j 6 k − 1. Their corresponding perpendicular subspaces (i.e.
hyperplanes) are
< X
16j6k−1
αjvj + vk>⊥=< vk+1, vk+2, . . . , vn, vj − αjvk|1 6 j 6 k − 1 > .
Proposition 5.3 Show that each of these hyperplanes uniquely determines a maximal clique of type I of the zero form.
Proof: When k = 1, v = v1 and < v >⊥=< v2, v3, . . . , vn>, it follows that its matrix representation is
lv(= l∞) = {
à 0 x2 x3 . . . xn−1 xn
−x2 0 0 . . . 0 0
!
|xi ∈ GF (q)}
consisting of those matrices whose first two rows as shown above, their first two columns obtained by skew-symmetry, and zero all other entries. Similar convention is used in the following.
When k > 2, the clique determined by v =P
16j6k−1(αjvj + vk) or its perpendic-ular space is denoted by lv = lα1,α2,...,αk−1 if there is no confusion. Let {e1, e2, . . . , en} be the standard basis of GF (q)n, i.e., ei = (0, 0, . . . , 0, 1, 0, 0, . . . , 0) with 1 in the i-th entry, 1 6 i 6 n.
Let
Li = {eTi · x − xT · ei|x ∈ GF (q)n with 0 in its i-th entry}, and,
α = (α1, α2, . . . , αi−1, 1, 0, 0, . . . , 0) ∈ GF (q)n
and Pi(α) denote the matrix obtained from the identity matrix by replacing its i-th row by the vector α. Then the matrix representation of lv = lα1,α2,...,αi−1, with respect to the fixed base {v1, v2, . . . , vn}, is given by
lv = lα1,α2,...,αi−1
= Pi(α)T · Li· Pi(α)
= {Pi(α)T·(ei·x−xT·ei)·Pi(α)|x ∈ GF (q)n with 0 in its i-th entry}
= {αT · x − xT · α|x ∈ GF (q)n with 0 in its i-th entry}.
Remark For attenuated space, i.e., Mk×n(GF (q)), those blocks of the zero matrix can be expressed as {αT · x|x ∈ GF (q)n} where α = (α1, α2, . . . , αi−1, 1, 0, 0, . . . , 0) ∈ GF (q)n for all nonzero α ∈ GF (q)n with 1 in its last nonzero entry.
For each i, there are qi−1 vectors of the form
α = (α1, α2, . . . , αi−1, 1, 0, 0, . . . , 0) ∈ GF (q)n,
and each such vector α uniquely determines a maximal clique lv = lα1,α2,...,αi−1 = Pi(α)T · Li· Pi(α) of zero form. Hence all maximal cliques of type I of the zero form are obtained in this way. The above observation can be summarized as follows:
Proposition 5.4
(1) Each maximal clique of type I consists of qn−1 vertices.
(2) Each vertex is in exactly qq−1n−1 maximal cliques of type I.
Proof: The maximal clique of type I consists of qn−1 vertices. since all maximal cliques
of type I is isomorphic to
uniquely determined by hyperplane H with dim(H) = n − 1. Then the number of hyperplanes is
Proposition 5.5 Those (qn − 1)/(q − 1) maximal cliques of type I containing the
zero form can be expressed as
{αT · x − xT · α|x = (x1, x2, . . . , xi−1, 0, xi+1, xi+2, . . . , xn), xj ∈ GF (q)n},
for α = (1, 0, 0, . . . , 0) or α = (α1, α2, . . . , αi−1, 1, 0, 0, . . . , 0) where i > 2 and αj ∈ GF (q)n.
Now, we turn to the intersection properties among those maximal cliques in the distance regular graph Alt(n, q). determine the rest q − 1 maximal cliques of type I are required. Q.E.D.
Let B be the set of all maximal cliques of type I in Alt(n, q). Then π = (An, B, ∈) turns out to be an incidence structure with some intersecting properties. Elements in B are called blocks of this incidence structure. The following proposition is simply a restatement of proposition 5.4 and lemma 5.6.
Proposition 5.7
(1) Each block B consists of qn−1 points,
(2) Each point is incident with exactly (qn− 1)/(q − 1) blocks,
(3) Any two distinct blocks are incident with either 0 or q common points,
(4) If B1, B2 ∈ B are distinct and B1∩ B2 6= φ then there are another q − 1 blocks, say B3, B4, . . ., Bq+1, such thatTq+1
i=1Bi = B1∩ B2 consists of q points.
Definition 5.8 A singular line is the intersection of two distinct maximal cliques.
When we only consider the maximal cliques of type I, we have the constant size of cliques. And if we define the graph with multiedge by mA,B = the number of singular lines containing vertices A and B, then we have mA,B = 0, 1 or q + 1.
Consider the definition of strongly regular multigraph, γ = P
y∈V mx,y(mx,y − 1).
For each A and B lie in the same maximal cliques, there are exactly 2 entries of B different from those of A. So we have γ =P
B(q + 1)q is the constant.
Definition 5.9 For a fixed matrix M(n−2)×(n−2)0 (GF (q)) with odd q, let Γ be a multi-graph with defined on (V, E) with {M| M is a skew-symmetric matrix over GF (q) satisfying M =
à X Y
−YT M0
!
} as the vertex set, and for any two vertices A, B, A ∼ B if and only if rank(A − B) = 2. Moreover, let mA,B be the number of singular line containing vertices A and B.
Theorem 5.10 Γ is a strongly regular multigraph with parameters
(m0, n0, µ0, γ0, R0) = (q + 1, qn−1, q2(q + 1), q(q2− 1), (q + 1)(qn−1− 1)),
and Γ is the collinearity graph of a 112-design with parameters
(r, k, t, c) = (q + 1, qn−1, q2, q(q − 1)).
Proof:
For a fixed vertex A ∈ V ,
(1) m0 = q + 1 is the number of maximal cliques contains A;
(2) there exists exactly q + 1 cliques containing A and B for adjacent vertices A and B, and each maximal clique has size qn−1; thus R0 = (q + 1)(qn−1− 1).
(3) γ0 =P
B∼AmA,B(mA,B− 1) =P
B∼A(q + 1)q = (q − 1)q(q + 1) = q(q2− 1), since mA,B = 1 or 0 whenever B does not lie in the q +1 maximal cliques which contains A.
(4) since m0(m0− n0) + µ0 = R0 + γ0, then (q + 1)(n0− (q + 1)) + µ0 =(q + 1)(qn−1− 1) + q(q2 − 1), it follows that µ0 = (q + 1)(qn−1− n0+ q2).
(5) since µ0v = (R0 + m0)(R0 + m0 − n0), then µ0q2n−3 = ((q + 1)(qn−1 − 1)) + (q + 1)((q + 1)(qn−1− 1) + (q + 1) − n0). It follows that µ0qn−2 = (q + 1)((q + 1)qn−1− n0);
combine (4) and (5), we have µ0qn−2 = qn−2(q + 1)(qn−1− n0 + q2). Hence n0 = qn−1 and µ0 = (q + 1)(qn−1− n0 + q2) = q2(q + 1).
We then have (r, k, t, c) = (q + 1, qn−1, q2, q(q − 1)) as required. Q.E.D.
The combinatorial interpretations of the parameters (r, k, t, c) of the 112-design un-der consiun-deration are given below:
(1) r = number of cliques contains a fixed vertex = q + 1.
(2) k = the clique size = qn−1.
(3) Fixed y ∈/ B where y is a point and B is a block, then t =P
x∈Bmx,y = q2. (4) Fixed y ∈ B, mx,y− 1 = q or 0 by x ∈ B and
a. mx,y = 1 if x does not lie in the q + 1 blocks containing y, b. mx,y = q + 1 if x lies in the q + 1 blocks containing y,
thus c =P
x∈B−{y}(mx,y− 1) = q(q − 1).
We constructed the strongly regular multigraph above. Now, we are interesting that in what conditions, we can make sure the 112-design with parameters
(r, k, t, c) = (q + 1, qn−1, q2, q(q − 1))
is the unique incidence structure such that the collinearity graph is this strongly regular multigraph with parameters
(m0, n0, µ0, γ0, R0) = (q + 1, qn−1, q2(q + 1), q(q2− 1), (q + 1)(qn−1− 1)).
We now check the numerical constraints required in Theorem 4.2.17 ([9], Theorem 4.4) for the uniqueness of the corresponding incidence structure with respect to the strongly regular multigraph under consideration; i.e., to find conditions to guarantee that m0 > 2, integral µ0 ≡ 0 mod m0, µ0 > 0, and n0 > max{m0− 1 + (µ0+mm020)γ0, 2(m0− 1)(µ0+ 1 − m0) + 2γ0,m0(m20−1)(µ0+ 1) + m0 γ20 + m0− 1}.
Theorem 5.11 The strongly regular multigraph Γ is the point graph of a unique 112-design whenever n = 6, q > 4 or n > 7, q > 3.
Proof: Clearly, µ0 = q2(q + 1) ≡ 0 mod m0(= q + 1) and µ0 > 0, m0 > 2.
(1) Since qn−1> q + (q2+ 1)q(q − 1) = q4− q3+ q2, then
qn−1 > (q + 1) − 1 + (q2(q + 1) + (q + 1))(q(q2 − 1))
(q + 1)2 ,
i.e., the condition n0 > m0− 1 + (µ0+mm020)γ0 hold.
(2) Since qn−1> 2q((q2− 1)(q + 1) + 1) + 2q(q2− 1) = 2q4+ 4q3− 2q2− 2q, then
qn−1 > 2((q + 1) − 1)(q2(q + 1) + 1 − (q + 1)) + 2q(q2 − 1),
i.e., the condition n0 > 2(m0− 1)(µ0 + 1 − m0) + 2γ0 hold.
(3) Since qn−1 > q(q+1)2 (q3+ q2+ 1) + (q + 1)q(q22−1) + q = 12(q5+ 3q4+ 2q3+ 2q), then
qn−1 > (q + 1)((q + 1) − 1)
2 (q2(q + 1) + 1) + (q + 1)q(q2− 1)
2 + (q + 1) − 1, i.e., the condition n0 > m0(m20−1)(µ0+ 1) + m0 γ20 + m0 − 1 hold.
Combine (1) ∼ (3), we have
qn−1 > max{q4− q3+ q2, 2q4+ 4q3− 2q2− 2q,1
2(q5+ 3q4 + 2q3+ 2q)} (*) (*) holds if n = 6, q > 4 and if n > 7, q > 3. Q.E.D.
From Theorem 5.10 and 5.11, we have:
Theorem 5.12 If Γ is a strongly regular multigraph with parameters (m0, n0, µ0, γ0, R0) = (q + 1, qn−1, q2(q + 1), q(q2− 1), (q + 1)(qn−1− 1)) with odd prime q and integer n such that n = 6, q > 4 or n > 7, q > 3, then Γ is the collinearity graph of a unique 112-design with parameters (r, k, t, c) = (q + 1, qn−1, q2, q(q − 1)).
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