Eigenvalue-free interval for threshold graphs

Linear Algebra and its Applications 583 (2019) 300–305

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Eigenvalue-free interval for threshold graphs Ebrahim Ghorbani a,b,∗ a

Department of Mathematics, K. N. Toosi University of Technology, P. O. Box 16765-3381, Tehran, Iran b School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 11 August 2019 Accepted 30 August 2019 Available online 4 September 2019 Submitted by R. Brualdi MSC: 05C50 05C75

a b s t r a c t This paper deals with the eigenvalues of the adjacency matrices of threshold graphs for which −1 and 0 are considered as trivial eigenvalues. We show that threshold √ have  graphs no non-trivial eigenvalues in the interval (−1 − 2)/2, √  (−1 + 2)/2 . This confirms a conjecture by Aguilar, Lee, Piato, and Schweitzer (2018). © 2019 Elsevier Inc. All rights reserved.

Keywords: Threshold graph Eigenvalue Anti-regular graph

1. Introduction A threshold graph is a graph that can be constructed from a one-vertex graph by repeated addition of a single isolated vertex to the graph, or addition of a single vertex that is adjacent to all other vertices. An equivalent definition is the following: a graph is a threshold graph if there are a real number S and for each vertex v a real weight w(v) * Correspondence to: Department of Mathematics, K. N. Toosi University of Technology, P. O. Box 16765-3381, Tehran, Iran. E-mail address: [email protected]. https://doi.org/10.1016/j.laa.2019.08.028 0024-3795/© 2019 Elsevier Inc. All rights reserved.

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such that two vertices u, v are adjacent if and only if w(u) + w(v) > S. This justifies the name “threshold graph” as S is the threshold for being adjacent. Threshold graphs also can be defined in terms of forbidden subgraphs, namely they are {P4 , 2K2 , C4 }-free graphs. Note, if a threshold graph is not connected then (since 2K2 is forbidden) at most one of its components is non-trivial (others are trivial, i.e. isolated vertices). For more information on properties of threshold graphs and related family of graphs see [5,10]. In this paper we deal with eigenvalues of the adjacency matrices of threshold graphs for which −1 and 0 are considered as trivial eigenvalues. In [9] it was shown that threshold graphs have no eigenvalues in (−1, 0). This result was extended in [7] by showing that a graph G is a cograph (i.e. a P4 -free graph) if and only if no induced subgraph of G has an eigenvalue in the interval (−1, 0). A distinguished subclass of threshold graphs is the family of anti-regular graphs which are the graphs with only two vertices of equal degrees. If G is anti-regular it follows easily that the complement graph G is also anti-regular. Up to isomorphism, there is only one connected anti-regular graph on n vertices and its complement is the unique disconnected n-vertex anti-regular graph [4]. In this paper we show that threshold graphs have no non-trivial eigenvalues in the √ √   interval (−1 − 2)/2, (−1 + 2)/2 . This result confirms a conjecture of [2] and improve the aforementioned result of [9]. We remark that another related conjecture was posed in [2] that for any n, the anti-regular graph with n vertices has the smallest positive eigenvalue and has the largest non-trivial negative eigenvalue among all threshold graphs on n vertices. Partial results on this conjecture is given in [1]. 2. Preliminaries In this section we introduce the notations and recall a basic fact which will be used in the sequel. The graphs we consider are all simple and undirected. For a graph G, we denote by V (G) the vertex set of G. For two vertices u, v, by u ∼ v we mean u and v are adjacent. If V (G) = {v1 , . . . , vn }, then the adjacency matrix of G is an n × n matrix A(G) whose (i, j)-entry is 1 if vi ∼ vj and 0 otherwise. By eigenvalues of G we mean those of A(G). The multiplicity of an eigenvalue λ of G is denoted by mult(λ, G). For a vertex v of G, let NG (v) denote the open neighborhood of v, i.e. the set of vertices of G adjacent to v and NG [v] = NG (v) ∪ {v} denote the closed neighborhood of v; we will drop the subscript G when it is clear from the context. Two vertices u and v of G are called duplicates if N (u) = N (v) and called coduplicates if N [u] = N [v]. Note that duplicate vertices cannot be adjacent while coduplicate vertices must be adjacent. A subset S of V (G) such that N (u) = N (v) for any u, v ∈ S is called a duplication class of G. Coduplication classes are defined analogously. We will make use of the interlacing property of graph eigenvalues which we recall below (see [6, Theorem 2.5.1]). Lemma 1. Let G be a graph of order n, H be an induced subgraph of G of order m, λ1 ≥ · · · ≥ λn and μ1 ≥ · · · ≥ μm be the eigenvalues of G and H, respectively. Then

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Fig. 1. A threshold graph: Vi ’s are cliques, Ui ’s are cocliques, each thick line indicates the edge set of a complete bipartite subgraph on some Ui , Vj .

λi ≥ μi ≥ λn−m+i for i = 1, . . . , m. In particular, if m = n − 1, then λ1 ≥ μ1 ≥ λ2 ≥ μ2 ≥ · · · ≥ λn−1 ≥ μn−1 ≥ λn . 3. Eigenvalue-free interval for threshold graphs In this section we present the main result of the paper. We start by the following remark on the structure of threshold graphs. Remark 2. As it was observed in [11] (see also [3,8]), the vertices of any connected threshold graph G can be partitioned into h non-empty coduplication classes V1 , . . . , Vh and h non-empty duplication classes U1 , . . . , Uh such that the vertices in V1 ∪ · · · ∪ Vh form a clique and N (u) = V1 ∪ · · · ∪ Vi for any u ∈ Ui , 1 ≤ i ≤ h. (It turns out that U1 ∪ · · · ∪ Uh form a coclique.) Accordingly, a connected threshold graph is also called nested split graph (or NSG for short). If mi = |Ui | and ni = |Vi | for 1 ≤ i ≤ h, then we write G = NSG(m1 , . . . , mh ; n1 , . . . , nh ). For an illustration of this structure with h = 5, see Fig. 1. It follows that a threshold graph G of order n is anti-regular if and only if n1 = · · · = nh = m1 = · · · = mh = 1 (in case n is even) or n1 = · · · = nh = m1 = · · · = mh−1 = 1 and mh = 2 (in case n is odd). In any graph G if we add a new vertex duplicate (coduplicate) to u ∈ V (G), then the multiplicity of 0 (of −1) increases by 1. That’s why the eigenvalues 0 and −1 are treated as trivial eigenvalues in threshold graphs. The following lemma can be deduced in a similar manner.

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Lemma 3. Let G = NSG(m1 , . . . , mh ; n1 , . . . , nh ) be a connected threshold graphs. Then mult(0, G) =

h  (mi − 1), i=1

h  mult(−1, G) = (ni − 1) + i=1

 1 0

if mh = 1, if mh ≥ 2.

Lemma 4. Let G be threshold graph which is not an anti-regular graph. Then there is some vertex v of G such that for H = G − v we have either (i) mult(0, G) = mult(0, H) + 1, mult(−1, G) = mult(−1, H); or (ii) mult(0, G) = mult(0, H), mult(−1, G) = mult(−1, H) + 1. Proof. Let G = NSG(m1 , . . . , mh ; n1 , . . . , nh ). First assume that n1 = · · · = nh = m1 = · · · = mh−1 = 1. As G is not an anti-regular graph, we have mh ≥ 3. Let v ∈ Uh and H = G − v. Then H = NSG(m1 , . . . , mh−1 , mh − 1; n1 , . . . , nh ). So by Lemma 3, mult(0, G) = 2 = mult(0, H) + 1, mult(−1, G) = 0 = mult(−1, H), and so we are done. So we may assume that nk ≥ 2 for some 1 ≤ k ≤ h or mj ≥ 2 for some 1 ≤ j ≤ h − 1. If nk ≥ 2, let v ∈ Vk and H = G − v. Then H = NSG(m1 , . . . , mh ; n1 , . . . , nk−1 , nk − 1, nk+1 , . . . , nh ). So by Lemma 3, mult(0, G) = mult(0, H), mult(−1, G) = mult(−1, H) + 1. If mj ≥ 2 for some 1 ≤ j ≤ h − 1, the result follows similarly.

2

By η+ (G) we denote the smallest positive eigenvalue of G and by η− (G) we denote the largest eigenvalue of G less than −1. The unique connected anti-regular graph on n ≥ 2 vertices is denoted by An . Lemma 5 ([2]). For anti-regular graphs we have η− (An ) < (−1 −

√

2)/2, and

(−1 +

√

2)/2 < η+ (An ).

We are now in a position to prove the main result of the paper. √  Theorem 6. Other than the trivial eigenvalues −1, 0, the interval (−1 − 2)/2, (−1 + √  2)/2 does not contain an eigenvalue of any threshold graph.

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Proof. Let G be a threshold graph of order n. We proceed by induction on n. With no loss of generality, we may assume that G is connected. The assertion holds if n ≤ 3, so we assume n ≥ 4. Recall that threshold graphs have no eigenvalues in the interval (−1, 0). If G is an anti-regular graph, then we are done by Lemma 5. So assume that G is not an anti-regular graph. Let v and H = G − v be as given in Lemma 4. Let λ1 ≥ · · · ≥ λn and μ1 ≥ · · · ≥ μn−1 be the eigenvalues of G and H, respectively. We may suppose that for some t, μt−−1 > μt− = · · · = μt−1 = 0 > μt = · · · = μt+j−1 = −1 > μt+j .

(1)

It is possible that either  = 0 (i.e. H has no 0 eigenvalue) or j = 0 (i.e. H has no −1 eigenvalue) but we have j +  ≥ 1. By the induction hypothesis, μt−−1

−1 + = η+ (H) > 2

√

2

and μt+j

−1 − = η− (H) < 2

√

2

.

By interlacing, from (1) we have λt− ≥ λt−+1 = · · · = λt−1 = 0 ≥ λt ≥ λt+1 = · · · = λt+j−1 = −1 ≥ λt+j , λt−−1 ≥ μt−−1 and μt+j ≥ λt+j+1 . As G is not an anti-regular graph, the case (i) or (ii) of Lemma 4 occurs. If the case (i) occurs, then mult(0, G) =  + 1 = mult(0, H) + 1, mult(−1, G) = j = mult(−1, H). This is only possible if λt− = λt = 0 and λt+j = −1 if j ≥ 1. If the case (ii) occurs, then mult(0, G) =  = mult(0, H), mult(−1, G) = j + 1 = mult(−1, H) + 1 which implies that λt = λt+j = −1 and λt− = 0 if  ≥ 1. So in any case, λt− , . . . , λt+j ∈ {−1, 0}. It turns out that √ −1 + 2 , η+ (G) = λt−−1 ≥ μt−−1 = η+ (H) > 2 √ −1 − 2 η− (G) = λt+j+1 ≤ μt+j = η− (H) < . 2 The result now follows. 2 Declaration of competing interest There is no conflict of interest. Acknowledgements The author would like to thank Cesar Aguilar for pointing out an error in an earlier version of the paper. The research of the author was in part supported by a grant from IPM (No. 98050211).

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