Spectral characterizations of two families of nearly complete bipartite graphs
Chia-an Liu and Chih-wen Weng
It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each of which is obtained from a complete bipartite graph by adding a vertex and an edge incident on the new vertex and an original vertex, which are not determined by their spectra.
AMS 2000 subject classifications:Primary 05C50; secondary 15A18.
Keywords and phrases: Bipartite graph, adjacency matrix, deter- mined by the spectrum (DS).
1. Introduction
The adjacency matrix A = (aij) of a simple graph G is a 0-1 square matrix with rows and columns indexed by the vertex set V (G) of G such that for any i, j∈ V (G), aij = 1 iff i, j are adjacent in G. The spectrum of G is the set of eigenvalues of its adjacency matrix A together with their multiplicities. Two graphs are cospectral (also known as isospectral) if they share the same graph spectrum. To start our study, let us consider the smallest non-isomorphic cospectral graphs first given by Cvetkovi´c [7] as shown in Figure1: the graph union K2,2∪ K1 and the star graph K1,4, where Kp,q denotes the complete bipartite graph of bipartition orders p and q. It is quick to check that their spectrum are both {[0]3,±2}. More constructions of cospectral graphs can be found in [16,8,18,21,10].
A graph G is determined by the spectrum if all the cospectral graphs of G are isomorphic to G. We abbreviate ‘determined by the spectrum’ to DS in the following. The question ‘which graphs are DS?’ goes back for more than half a century and originates from chemistry [17, 6], [9, Chapter 6]. After that, there appeared many examples and applications for the DS graphs.
One of them is that in 1966 Fisher [15] modeled the shape of a drum by a arXiv:1601.07012v1
241
Figure 1: Two non-isomorphic cospectral graphs K2,2∪ K1 and K1,4.
graph from considering a question of Kac [19]: ‘Can you hear the shape of a drum?’ To see more details, Van Dam, Haemers and Brouwer gave a great amount of surveys for the DS graphs [10,11,12], [2, Chapter 14] in the past decades.
In [13] the non-regular bipartite graphs with four distinct eigenvalues were studied, and whether a such connected graph on at most 60 vertices is DS or not was determined. In [14] bipartite biregular graphs with 5 eigen- values were studied, and all such connected graphs on at most 33 vertices were determined. In this research we study two families of nearly complete bipartite graphs one-edge different from a complete bipartite graph which also have 4 or 5 distinct eigenvalues without the assumptions of regularity, connectivity, or bounds on the number of vertices.
Let G be a simple bipartite graph with e edges. The spectral radius ρ(G) of G is the largest eigenvalue of the adjacency matrix of G. It was shown in [1, Proposition 2.1] that ρ(G)≤√
e with equality if and only if G is a complete bipartite graph with possibly some isolated vertices. It is straightforward to show that for any positive integer p the regular complete bipartite graph Kp,p is DS but, for example, the non-isomorphic bipartite graphs K1,6 and K2,3∪ 2K1 are cospectral. There are several results extending [1, 4, 20, 5]
of the above bound, which aim to solve an analog of the Brualdi-Hoffman conjecture for non-bipartite graphs [3], proposed in [1].
Our research is motivated from the following twin primes bound proposed in [5, Theorem 5.2]: For e ≥ 4, (e − 1, e + 1) is a pair of twin primes if and only if
ρ(e) <
e +
e2− 4(e − 1 −√ e− 1) 2
where ρ(e) denotes the maximal ρ(G) of a bipartite graph G on e edges which is not a union of a complete bipartite graph and possibly some isolated ver- tices. We need to introduce the notations Kp,q− and Kp,q+ of the graphs which
Figure 2: The graphs K3,4− and K3,4+ which are one-edge different from K3,4.
are one-edge different from a complete bipartite graph. For 2 ≤ min{p, q}, let Kp,q− denote the graph with pq− 1 edges obtained from Kp,q by deleting an edge, and Kp,q+ denote the graph with pq + 1 edges obtained from Kp,qby adding a new vertex x and a new edge xy where y is a vertex in the partite set of order min{p, q}. Note that K2,q+ = K2,q+1− for q ≥ 2. Two examples of such graphs are shown in Figure2.
The paper is organized as follows. Preliminary results are in Section2.
Theorem10 in Section 3 proves that all the graphs Kp,q− for 2≤ p ≤ q are DS. Then Theorem 13 in Section 4 find all the pairs (p, q) such that the bipartite graph Kp,q+ is DS. Furthermore, for each Kp,q+ which is not DS we also find its unique non-isomorphic cospectral graph.
2. Preliminary
Basic results are provided in this section for later use.
Lemma 1. ([1, Proposition 2.1]) Let G be a simple bipartite graph with e edges. Then
ρ(G)≤√ e
with equality iff G is a disjoint union of a complete bipartite graph and isolated vertices.
The following result gives the relations between the spectrum and the numbers of vertices and edges in a graph which is proved simply by the definition of the adjacency matrix and its square.
Proposition 2. Let G be a simple graph with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn. Then
(i) G has n vertices, and
(ii) G has 12
n i=1
λ2i edges.
Since we focus on the bipartite graphs, a well-known spectral character- ization of bipartite graphs [2, Proposition 3.4.1] is used in this paper.
Proposition 3. A simple graph G is bipartite if and only if for each eigen- value λ of G, −λ is also an eigenvalue of G with the same multiplicity.
Then a spectral characterization of complete bipartite graphs is straight- forward.
Proposition 4. Let G be a simple graph with spectrum {[0]n−2,±λ} where n≥ 2 is the number of vertices in G. Then λ2 is a nonnegative integer, and G is the union of some isolated vertices (if any) and a complete bipartite graph with λ2 edges.
Proof. By Proposition3and Proposition2(ii), G is bipartite with λ2 edges.
Since the equality holds in Lemma 1, the completeness follows.
From Proposition4one can quickly find all the complete bipartite graphs which are DS.
Corollary 5. For any positive integers p ≤ q, Kp,q is DS if and only if p≤ p and q ≥ q for any positive integers p≤ q satisfying pq = pq.
It is not difficult to compute the spectrum of each bipartite graph Kp,q− or Kp,q+ [4,20,5].
Proposition 6. Let 2≤ min{p, q} be positive integers.
(i) The graph Kp,q− has spectrum
⎧⎨
⎩[0]p+q−4,±
pq− 1 ±
(pq− 1)2− 4(p − 1)(q − 1) 2
⎫⎬
⎭, and
(ii) the graph Kp,q+ has spectrum
⎧⎨
⎩[0]p+q−3,±
pq + 1±
(pq + 1)2− 4(p − 1)q 2
⎫⎬
⎭.
We introduce some sets of bipartite graphs used in Lemma 8 and latter content.
Definition 7. Define the sets of bipartite graphs K0 := {Kp,q | p, q ∈N},
K− := {Kp,q− | 2 ≤ p ≤ q, (p, q) = (2, 2)}, K+ := {Kp,q+ | 2 ≤ p ≤ q}, and
K := K0∪K−∪K+.
Lemma 8. ([5, Lemma 4.1]) Let G be a simple bipartite graph on e edges without isolated vertices. If the spectral radius ρ(G) of G satisfies
ρ(G)≥
e +
e2− 4(e − 1 −√ e− 1)
2 ,
then G∈K.
The following result [11, Proposition 1] is well-known.
Proposition 9. The path with n vertices is DS.
3. Spectral characterizations of Kp,q−
Note that the set K− of nearly bipartite graphs is defined in Definition 7.
We prove that each graph G∈ {K2,2− } ∪K− is DS in this section.
Theorem 10. For any positive integers 2≤ p ≤ q, the graph Kp,q− is DS.
Proof. If p = q = 2 then Kp,q− is a path on 4 vertices. Hence K2,2− is DS by Proposition9. Let 2 < q and G be a simple graph with the same spectrum as Kp,q−. From Proposition 2, the numbers of vertices and edges in G are
|V (G)| = p + q and |E(G)| = pq − 1. Additionally, Proposition 3 tells that G is a bipartite graph.
Suppose G has at least 2 nontrivial components G1 and G2, where a nontrivial component is a connected graph with at least one edge. Then the spectra of G1and G2share the nonzero eigenvalues of G. Since G is bipartite, G1 and G2 are both bipartite. By Proposition 3 again and without loss of generality we have sp(G1) = {[0]m1,±e1} and sp(G2) = {[0]m2,±e2} for some nonnegative integers m1, m2 with m1+ m2+ 4≤ p + q, where
e1=
pq− 1 +
(pq− 1)2− 4(p − 1)(q − 1) 2
and
e2 =
pq− 1 −
(pq− 1)2− 4(p − 1)(q − 1) 2
by Proposition 6 (i). From Proposition 4 G1 is a complete bipartite graph with e21 edges, and thus (pq− 1)2− 4(p − 1)(q − 1) is a perfect square of type (pq− 1 − 2k)2 for some k∈N. However,
(pq− 1)2− 4(p − 1)(q − 1) = (pq − 1)2− 4(pq − p − q + 1)
> (pq− 1)2− 4(pq − 2)
= (pq− 3)2, which is a contradiction.
Therefore G has exactly one nontrivial component G0. Then
sp(G0) =
⎧⎨
⎩[0]m,±
pq− 1 ±
(pq− 1)2− 4(p − 1)(q − 1) 2
⎫⎬
⎭
for some nonnegative integer m with
(1) |V (G0)| = m + 4 ≤ p + q.
Then by Proposition 2 (ii)
(2) e :=|E(G0)| = |E(G)| = pq − 1.
Note that the spectral radius of G0
ρ(G0) =
pq− 1 +
(pq− 1)2− 4(p − 1)(q − 1) 2
=
e +
e2− 4(e − 1 − (p + q − 3)) 2
≥
e +
e2− 4(e − 1 −√ e− 1)
2 ,
since
(p + q− 3)2− (e − 1) = (p − 2)2+ (q− 3)2+ p(q− 2) − 2 ≥ 0
for 2≤ p ≤ q and 3 ≤ q. By Lemma8G0∈K. From Proposition4G0is not a complete bipartite graph. Hence G0 ∈K−or G0 ∈K+. Suppose G0∈K+, i.e., G0 = Kp+,q for some 2≤ p≤ q. Then by (1)
(3) |V (G0)| = p+ q+ 1≤ p + q, and by (2)
(4) |E(G0)| = pq+ 1 = e = pq− 1.
According to Proposition 6(ii),
(5) (p− 1)q = (p− 1)(q − 1).
(4) and (5) imply q + 3 = p + q. Then by (3) p ≤ 2, and hence p = 2.
Therefore G0 = K2,q+ = K2,q−+1 for some q ≥ 2, and we have G0 ∈ K−. Let G0 = Kp−,q for some 2 ≤ p ≤ q and 3 ≤ q. Then we rewrite the equations (3), (4) and (5) as
|V (G0)| = p+ q≤ p + q, (6)
|E(G0)| = pq− 1 = pq − 1 and (7)
(p− 1)(q− 1) = (p − 1)(q − 1) (8)
respectively, where the third equation (8) is from Proposition6 (i). (7) and (8) imply|V (G0)| = p+ q= p + q =|V (G)|. Hence G0 = G. The equalities in both sum and product of p≤ q and p≤ q imply that (p, q) = (p, q).
Hence G = G0 = Kp,q− , and the result follows.
Remark 11. From Theorem 10 we have K2,q+ = K2,q+1− is DS for 2 ≤ q.
However, not all graphs in K+ are DS. For example, the non-isomorphic graphs
Km+2,4m+2+ and K2m+2,2m+3− ∪ mK1
are cospectral for each m∈N.
Corollary 12. K2,q+ is DS for 2≤ q.
4. Spectral characterizations of Kp,q+
Theorem 10 shows that for 2 ≤ p ≤ q the graph Kp,q− is DS and its proof appears useful for the study on the graph Kp,q+ . However, Remark11imme- diately gives a family of Kp,q+’s which are not DS. In this section we present a
sufficient and necessary condition to determine whether Kp,q+ is DS or not for each pair (p, q). Furthermore, we also find all the non-isomorphic cospectral graphs of every Kp,q+ which is not DS.
Theorem 13. Let 3 ≤ p ≤ q be positive integers. Then Kp,q+ is not DS if and only if the quadratic polynomial
(9) x2− (q + 3)x + (pq + 2) = 0
has two integral roots p ≤ q in [2,∞). Moreover, if Kp,q+ is not DS then Kp−,q∪ (p − 2)K1 is its unique non-isomorphic cospectral graph.
Proof. Let G be a simple graph with the same spectrum as Kp,q+ . We prove Theorem13by two steps. In the first part of proof we show that G0∈K−∪ K+where G0is obtained from G by deleting all the isolated vertices (if any).
This process is similar to what we have done in the proof of Theorem 10. In the second part of proof, we prove that Kp,q+ is not DS if and only if (9) has two integral roots p, q and G0 = Kp−,q other than Kp,q+ itself. Moreover, if Kp,q+ is not DS then its only non-isomorphic cospectral graphs is obtained by adding p− 2 many isolated vertices to Kp−,q.
From Proposition2, the numbers of vertices and edges in G are|V (G)| = p + q + 1 and |E(G)| = pq + 1. Additionally, Proposition 3tells that G is a bipartite graph. Suppose G has at least 2 nontrivial components G1 and G2. Then the spectra of G1 and G2 share the nonzero eigenvalues of G. Since G is bipartite, G1 and G2 are both bipartite. By Proposition3and without loss of generality, we have sp(G1) ={[0]m1,±e1} and sp(G2) ={[0]m2,±e2} for some nonnegative integers m1, m2 with m1+ m2+ 4≤ p + q + 1, where
e1=
pq + 1 +
(pq + 1)2− 4(p − 1)q 2
and
e2=
pq + 1−
(pq + 1)2− 4(p − 1)q 2
by Proposition 6 (ii). From Proposition 4, G1 is a complete bipartite graph with e21 edges, and thus (pq + 1)2 − 4(p − 1)q is a perfect square of type (pq + 1− 2k)2 for some k∈N. However,
(pq + 1)2− 4(p − 1)q > (pq + 1)2− 4pq = (pq − 1)2, which is a contradiction.
Therefore G has exactly one nontrivial component G0. Then
sp(G0) =
⎧⎨
⎩[0]m,±
pq + 1±
(pq + 1)2− 4(p − 1)q 2
⎫⎬
⎭
for some nonnegative integer m with
(10) |V (G0)| = m + 4 ≤ p + q + 1.
Then by Proposition2 (ii)
(11) e :=|E(G0)| = |E(G)| = pq + 1.
Note that the spectral radius of G0
ρ(G0) =
pq + 1 +
(pq + 1)2− 4(p − 1)q 2
=
e +
e2− 4(e − 1 − q) 2
≥
e +
e2− 4(e − 1 −√ e− 1)
2 ,
since
q2− (e − 1) = q2− pq = q(q − p) ≥ 0
for 3 ≤ p ≤ q. By Lemma 8 G0 ∈ K. From Proposition 4, G0 is not a complete bipartite graph. Hence G0 ∈ K− or G0 ∈K+. Here we complete the first part of proof.
Suppose G0∈K+, i.e., G0= Kp+,q for some 2≤ p ≤ q. Then by (10) (12) |V (G0)| = p+ q+ 1≤ p + q + 1,
and by (11)
(13) |E(G0)| = pq+ 1 = e = pq + 1.
According to Proposition 6(ii),
(14) (p− 1)q= (p− 1)q.
(13) and (14) imply q = q and p= p. Therefore G = G0= Kp,q+.
Suppose G0 ∈K−. Let G0 = Kp−,q for some 2 ≤ p ≤ q and 3≤ q. Similar to the equations (12), (13) and (14) above, G is not DS if and only if there exists integral pair (p, q) that satisfies
|V (G0)| = p+ q≤ p + q + 1, (15)
|E(G0)| = pq− 1 = pq + 1 and (16)
(p− 1)(q− 1) = (p − 1)q, (17)
where the third equation (17) is from Proposition6. Note that (16) and (17) imply
(18) p+ q= q + 3
and hence (15) automatically holds. Conversely, (16) and (18) imply (17).
Hence the graph G0 = Kp−,q exists if and only if quadratic polynomial in (9) has two integral roots p, q. Note that G0 = Kp−,q is the only graph found in K−∪K+ except for Kp,q+ . Hence we conclude that for each pair of positive integers 3≤ p ≤ q, Kp,q+ is not DS if and only if (9) has two integral roots p, q and the only non-isomorphic cospectral graph is obtained from Kp−,q by adding (p + q + 1)− (p+ q) = p− 2 many isolated vertices by (18). Here we complete the second part of proof, and the result follows.
The following lemma helps us to exhaustedly enumerate Kp,q+ which has a non-isomorphic cospectral graph by using Theorem 13.
Lemma 14. Let 3≤ p ≤ q be integers. Then the quadratic polynomial in (9) has two integral roots p, q ∈ [2, ∞) if and only if there exist nonnegative integers a, b, b, t with 1 ≤ b, b < a, gcd(a, b) = 1, bb ≡ 1 (mod a), and bb+ t≥ 2 such that p, q, p, q can be written as
p = b(a− b)t + bb+b(1− bb) a + 1, (19)
q = a2t + ab, (20)
p = bq/a + 1, and q= q + 2− bq/a.
(21)
Proof. For the necessity, suppose that the quadratic polynomial in (9) has two integral roots p, q∈ [2, ∞). Then pq= pq + 2 and p+ q= q + 3.
Thus
(22) (p− 1) · (q + 1 − (p− 1)) = (p − 1)q.
Note that p− 1 ≤ q, otherwise we have p = 2 which contradicts to 3 ≤ p.
Let
(23) p− 1 = bq
a
where 1 ≤ b < a are integers and gcd(a, b) = 1. Then q ≡ 0 (mod a). Let q = as for some s∈N. Thus (22) becomes
(24) b((a− b)s + 1)
a = p− 1.
Then (a− b)s + 1 ≡ 0 (mod a) since gcd(a, b) = 1. Hence bs ≡ 1 (mod a). Let s = at + b where t and 1 ≤ b < a are nonnegative integers with bb = bs− bat ≡ 1 (mod a). Therefore, q = as = a2t + ab as stated in (20).
Substituting the above s = at + b into (24), we have (19). The formulae of p and q are immediate from (23) and (18). Note that if t = 0 and bb = 1 then p = 2, violating the assumption p≥ 3. Hence bb+ t≥ 2.
For the sufficiency, we check that for nonnegative integers a, b, b, t sat- isfying 1 ≤ b, b < a, gcd(a, b) = 1, bb ≡ 1 (mod a), and bb+ t ≥ 2, the corresponding values of p, q, p, qare feasible. Note that we can rewrite (19) to (21) as
p = b[(a− b)(at + b) + 1]
a + 1,
q = a(at + b),
p = b(at + b) + 1, and q = (at + b)(a− b) + 2.
One can immediately see that p, q are both integers not less than 2. More- over, the sum and product of p, q are
p+ q= q + 3 pq = pq + 2 ,
which imply that p, q are the two integral roots of (9).
To quickly find a non-isomorphic cospectral graphs pair which are nearly complete bipartite, a special case of Theorem13is provided in the following corollary.
Corollary 15. For each pair of positive integers (t, a) with a≥ 2, the graph K(a+−1)t+2,a2t+a
is not DS. Moreover,
Kat+2,a(a− −1)t+a+1∪ (a − 1)tK1
is its unique cospectral graph.
Proof. Let b = b = 1 in Lemma14. Then (19) to (21) become that p = (a− 1)t + 2,
q = a2t + a,
p = at + 2, and q= a(a− 1)t + a + 1.
Substituting these data into Theorem13, we immediately have the proof.
Example 16. By computer program, we list all Kp,q+’s that are not DS for q ≤ 20 in the following table including the corresponding unique cospec- tral graphs and the values of parameters a, b, b, t. Note that the choices of (a, b, b, t) are not unique.
The unique cospectral graph (a, b, b, t) K3,6+ K4,5− ∪ K1 (2, 1, 1, 1) or (3, 2, 2, 0) K4,10+ K6,7− ∪ 2K1 (2, 1, 1, 2) or (5, 3, 2, 0) K5,14+ K8,9− ∪ 3K1 (2, 1, 1, 3) or (7, 4, 2, 0) K4,12+ K5,10− ∪ 2K1 (3, 1, 1, 1) or (4, 3, 3, 0) K5,15+ K7,11− ∪ 3K1 (3, 2, 2, 1) or (5, 2, 3, 0) K6,18+ K10,11− ∪ 4K1 (2, 1, 1, 4) or (9, 5, 2, 0) K5,20+ K6,17− ∪ 3K1 (4, 1, 1, 1) or (5, 4, 4, 0)
Acknowledgements
This research is supported by the National Science Council of Taiwan R.O.C.
and Ministry of Science and Technology of Taiwan R.O.C. respectively under the projects NSC 102-2115-M-009-009-MY3 and MOST 103-2632-M-214- 001-MY3-2 (including its subproject MOST 104-2811-M-214-001).
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Chia-an Liu
Department of Financial and Computational Mathematics I-Shou University
Kaohsiung Taiwan
E-mail address: [email protected]
Chih-wen Weng
Department of Applied Mathematics National Chiao-Tung University Hsinchu
Taiwan
E-mail address: [email protected] Received March 3, 2016
