Linear Algebra and its Applications 581 (2019) 336–353

Contents lists available at ScienceDirect

Linear Algebra and its Applications

www.elsevier.com/locate/laa

Signless Laplacian eigenvalue problems

of Nordhaus–Gaddum type

Xueyi Huang a, Huiqiu Lin b,∗

a School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, PR Chinab Department of Mathematics, East China University of Science and Technology, Shanghai 200237, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 May 2019Accepted 19 July 2019Available online 25 July 2019Submitted by R. Brualdi

MSC:05C50

Keywords:Signless Laplacian eigenvalueNordhaus–Gaddum type inequalitiesInterlacingQuotient matrix

Let G be a graph of order n, and let q1(G) ≥ q2(G) ≥ · · · ≥qn(G) denote the signless Laplacian eigenvalues of G. Ashraf and Tayfeh-Rezaie (2014) [3] showed that q1(G) + q1(G) ≤3n − 4, with equality holding if and only if G or G is the star K1,n−1. In this paper, we prove that q2(G) + q2(G) ≤ 2n − 4, where the equality holds if and only if G or G is K2, P4 or C4. Also, we discuss the following problem: for n ≥ 6, does q2(G) +q2(G) ≤ 2n − 5 always hold? We provide positive answers to this problem for the graphs with disconnected complements and the bipartite graphs, and determine the graphs attaining the bound. Moreover, we show that q2(G) + q2(G) ≥ n − 2, and the extremal graphs are also characterized.

© 2019 Elsevier Inc. All rights reserved.

1. Introduction

Let G be a simple graph with vertex set V (G) = {v1, . . . , vn}. We denote the comple-ment of G by G, the adjacency matrix of G by A(G), and the degree (resp. neighborhood)

* Corresponding author.E-mail addresses: huangxy@zzu.edu.cn (X. Huang), huiqiulin@126.com (H. Lin).

https://doi.org/10.1016/j.laa.2019.07.0260024-3795/© 2019 Elsevier Inc. All rights reserved.

The second eigenvalue of some normal Cayley

graphs of highly transitive groups

Xueyi Huang∗

School of Mathematics and StatisticsZhengzhou University

Zhengzhou, China

huangxymath@gmail.com

Qiongxiang Huang†

College of Mathematics and Systems ScienceXinjiang University

Urumqi, China

huangqx@xju.edu.cn

Sebastian M. Cioaba‡

Department of Mathematical SciencesUniversity of Delaware

Newark, USA

cioaba@udel.edu

Submitted: Aug 3, 2018; Accepted: May 21, 2019; Published: Jun 21, 2019

c©The authors. Released under the CC BY-ND license (International 4.0).

Abstract

Let G be a finite group acting transitively on [n] = {1, 2, . . . , n}, and let Γ =Cay(G,T ) be a Cayley graph of G. The graph Γ is called normal if T is closedunder conjugation. In this paper, we obtain an upper bound for the second (largest)eigenvalue of the adjacency matrix of the graph Γ in terms of the second eigenvaluesof certain subgraphs of Γ. Using this result, we develop a recursive method todetermine the second eigenvalues of certain Cayley graphs of Sn, and we determinethe second eigenvalues of a majority of the connected normal Cayley graphs (andsome of their subgraphs) of Sn with maxτ∈T |supp(τ)| 6 5, where supp(τ) is the setof points in [n] non-fixed by τ .

Mathematics Subject Classifications: 05C50

1 Introduction

Let Γ = (V (Γ), E(Γ)) be a simple undirected graph of order n with adjacency matrixA(Γ). The eigenvalues of A(Γ), denoted by λ1(Γ) > λ2(Γ) > · · · > λn(Γ), are also called

∗Supported by the China Postdoctoral Science Foundation under grant 2019M652556 and the Post-doctoral Research Sponsorship in Henan Province under grant 1902011.†Corresponding author. Supported by the NSFC grants 11531011 and 11671344.‡Supported by the NSF grants DMS-1600768 and CIF-1815922.

the electronic journal of combinatorics 26(2) (2019), #P2.44 1

Journal of Algebraic Combinatorics (2019) 50:99–111https://doi.org/10.1007/s10801-018-0843-1

The second largest eigenvalues of some Cayley graphs onalternating groups

Xueyi Huang1 ·Qiongxiang Huang1

Received: 13 December 2017 / Accepted: 21 September 2018 / Published online: 15 October 2018© Springer Science+Business Media, LLC, part of Springer Nature 2018

AbstractLet An denote the alternatinggroupof degreenwithn ≥ 3.The alternatinggroupgraphAGn , extended alternating group graph E AGn and complete alternating group graphCAGn are the Cayley graphs Cay(An, T1), Cay(An, T2) and Cay(An, T3), respec-tively, where T1 = {(1, 2, i), (1, i, 2) | 3 ≤ i ≤ n}, T2 = {(1, i, j), (1, j, i) | 2 ≤i < j ≤ n} and T3 = {(i, j, k), (i, k, j) | 1 ≤ i < j < k ≤ n}. In this paper, wedetermine the second largest eigenvalues of AGn , E AGn and CAGn .

Keywords Alternating group graph · Cayley graph · Second largest eigenvalue

Mathematics Subject Classification 05C50

1 Introduction

LetG = (V (G), E(G)) be a simple undirected graph of order n. The adjacencymatrixof G, denoted by A(G), is the n × n matrix with entries auv = 1 if {u, v} ∈ E(G)

and auv = 0 otherwise. The eigenvalues of A(G) are denoted by λ1(G) ≥ λ2(G) ≥· · · ≥ λn(G), which are also called the eigenvalues of G.

For v ∈ V (G), we denote by N (v) = {u ∈ V (G) | {u, v} ∈ E(G)} and d(v) =|N (v)| the neighborhood and degree of v, respectively. Let D(G) = diag(d(v) | v ∈V (G)) denote the diagonal degree matrix of G. The Laplacian matrix of G is definedas L(G) = D(G) − A(G), which is positive semi-definite and always has 0 as its

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11531011,11671344).

B Qiongxiang Huanghuangqx@xju.edu.cn

Xueyi Huanghuangxymath@gmail.com

1 College of Mathematics and Systems Science, Xinjiang University, Ürümqi 830046, Xinjiang,People’s Republic of China

123

AlgebraColloquiumc⃝ 2019 AMSS CAS

& SUZHOU UNIV

Algebra Colloquium 26 : 1 (2019) 65–82

DOI: 10.1142/S1005386719000075

On Graphs with Three or Four Distinct Normalized

Laplacian Eigenvalues∗

Xueyi HuangSchool of Mathematics and Statistics, Zhengzhou University

Zhengzhou 450001, China

E-mail: huangxy@zzu.edu.cn

Qiongxiang Huang†

College of Mathematics and Systems Science, Xinjiang UniversityUrumqi 830046, China

E-mail: huangqx@xju.edu.cn

Received 16 November 2017Revised 15 September 2018

Communicated by Genghua Fan

Abstract. We characterize all connected graphs with exactly three distinct normalizedLaplacian eigenvalues among which one is equal to 1, and determine all connected bipar-tite graphs with at least one vertex of degree 1 having exactly four distinct normalizedLaplacian eigenvalues. In addition, we find all unicyclic graphs with three or four distinctnormalized Laplacian eigenvalues.

2010 Mathematics Subject Classification: 05C50

Keywords: normalized Laplacian eigenvalue, bipartite graph, symmetric BIBD, unicyclicgraph, Hadamard matrix

1 Introduction

Let G be a simple undirected graph on n vertices with m edges. The adjacencymatrix A = (auv) of G is the n × n matrix with rows and columns indexed bythe vertices, where auv = 1 if u is adjacent to v, and 0 otherwise. We denoteby D = diag(dv1 , dv2 , . . . , dvn) the diagonal degree matrix of G. The well-knownLaplacian matrix of G is defined as L = D − A. The normalized Laplacian matrix

∗This work is supported by the National Natural Science Foundation of China (grants No.11671344, 11531011 and 11701492).†Corresponding author.

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May 16, 2018 18:18 WSPC/S0219-4988 171-JAA 1950075

Journal of Algebra and Its Applications(2019) 1950075 (16 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0219498819500750

Enumerating Cayley (di-)graphs on dihedral groups

Xueyi Huang∗ and Qiongxiang Huang†

College of Mathematics and Systems ScienceXinjiang University

Urumqi, Xinjiang 830046, P. R. China∗huangxymath@gmail.com

†huangqx@xju.edu.cn

Received 26 November 2017Accepted 2 April 2018Published 18 May 2018

Communicated by Tai Huy Ha

Let p be an odd prime, and D2p = 〈τ, σ | τp = σ2 = e, στσ = τ−1〉 the dihedral groupof order 2p. In this paper, we provide the number of (connected) Cayley (di-)graphs onD2p up to isomorphism by using the Polya enumeration theorem. In the process, we alsoenumerate (connected) Cayley digraphs on D2p of out-degree k up to isomorphism foreach k.

Keywords: Cayley (di-)graph; dihedral group; Cayley isomorphism; Polya enumerationtheorem.

Mathematics Subject Classification: 05C25

1. Introduction

Let G be a finite group, and let S be a subset of G such that e �∈ S. The Cayleydigraph on G with respect to S, denoted by Cay(G,S), is the digraph with vertexset G and with an arc from g to h if hg−1 ∈ S. If S is symmetric, i.e. S−1 ={s−1 | s ∈ S} = S, then hg−1 ∈ S if and only if gh−1 ∈ S, and so Cay(G,S) can beviewed as an undirected graph, which is called the Cayley graph on G with respectto S. In particular, if G is a cyclic group, then the Cayley (di-)graph Cay(G,S) iscalled a circulant (di-)graph.

Let Cay(G,S) be the Cayley digraph on G with respect to S. Suppose thatα ∈ Aut(G), where Aut(G) is the automorphism group of G. Let T = α(S). Thenit is easily shown that α induces an isomorphism from Cay(G,S) to Cay(G, T ).Such an isomorphism is called a Cayley isomorphism. However, it is possible for

†Corresponding author.

1950075-1

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CC Creative Commons AMC Logo

Also available at http://amc-journal.eu

ARS MATHEMATICA CONTEMPORANEA x (xxxx) 1–x

On graphs with exactly two positive eigenvalues

Fang DuanCollege of Mathematics and Systems Science, Xinjiang University, Urumqi, P. R. China

School of Mathematics Science, Xinjiang Normal University, Urumqi, P. R. China

Qiongxiang Huang ∗

College of Mathematics and Systems Science, Xinjiang University, Urumqi, P. R. China

Xueyi Huang †

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, P. R. China

Received dd mmmm yyyy, accepted dd mmmmm yyyy, published online dd mmmmm yyyy

Abstract

The inertia of a graph G is defined to be the triplet In(G) = (p(G), n(G), η(G)),where p(G), n(G) and η(G) are the numbers of positive, negative and zero eigenvalues(including multiplicities) of the adjacency matrix A(G), respectively. Traditionally p(G)(resp. n(G)) is called the positive (resp. negative) inertia index of G. In this paper, weintroduce three types of congruent transformations for graphs that keep the positive inertiaindex and negative inertia index. By using these congruent transformations, we determineall graphs with exactly two positive eigenvalues and one zero eigenvalue.

Keywords: Congruent transformation, positive (negative) inertia index, nullity.

Math. Subj. Class.: 05C50

1 IntroductionAll graphs considered here are undirected and simple. For a graph G, let V (G) and E(G)denote the vertex set and edge set of G, respectively. The order of G is the number ofvertices of G, denoted by |G|. For v ∈ V (G), we denote by NG(v) = {u ∈ V (G) | uv ∈E(G)} the neighborhood of v, NG[v] = NG(v) ∪ {v} the closed neighborhood of v and

∗Corresponding author. Supported by the National Natural Science Foundation of China (Nos. 11671344,11531011).†Supported by the China Postdoctoral Science Foundation (No. 2019M652556), and the Postdoctoral Re-

search Sponsorship in Henan Province (No. 1902011).E-mail addresses: fangbing327@126.com (Fang Duan), huangqx@xju.edu.cn (Qiongxiang Huang),

huangxy@zzu.edu.cn (Xueyi Huang)

CC This work is licensed under http://creativecommons.org/licenses/by/3.0/

On graphs whose smallest distance (signless Laplacian)eigenvalue has large multiplicity

Lu Lu, Qiongxiang Huang and Xueyi Huang

College of Mathematics and Systems Science, Xinjiang University, Urumqi, P.R. China

ABSTRACT

Denote by ∂1 ≥ ∂2 ≥ · · · ≥ ∂n (resp. ∂Q1 ≥ ∂

Q2 ≥ · · · ≥ ∂

Qn ) the

distance (resp. distance signless Laplacian) eigenvaluesof a connectedgraph G on n vertices, and by m(∂i) (resp. m(∂

Qi )) the multiplicity of

∂i (resp. ∂Qi ). In this paper, we completely determine the graphs with

m(∂n) = n − 3 andm(∂Qn ) = n − 2, respectively.

ARTICLE HISTORYReceived 8 May 2017Accepted 25 September2017

COMMUNICATED BYS. Cioaba

KEYWORDSDistance matrix; distancesignless Laplacian matrix;smallest eigenvalue

AMS SUBJECTCLASSIFICATIONS05C50; 05C12; 15A18

1. Introduction

Let G = (V ,E) be a connected simple graph with vertex set V = {v1, v2, . . . , vn} and edgeset E = {e1, e2, . . . , em}. We denoted by dG(vi, vj) the distance between vi and vj, which isdefined as the length of a shortest path between them. The diameter ofG, denoted by d(G),is themaximum distance between any two vertices ofG. The distance matrix ofG, denotedby D(G), is the n × n matrix whose (i, j)-entry is equal to dG(vi, vj), for 1 ≤ i, j ≤ n.The transmission Tr(vi) of a vertex vi is defined as the sum of the distances between viand all other vertices in G, that is, Tr(vi) = ∑n

j=1 dG(vi, vj). For more details about thedistance matrix, we refer the reader to [1]. As the distance matrix D(G) is symmetric, allof its eigenvalues are real, we can list them as ∂1(G) ≥ ∂2(G) ≥ · · · ≥ ∂n(G). Denote bym(∂i(G)) themultiplicity of ∂i(G). The multi-set of these eigenvalues is called the distancespectrum of G, denoted by SpecD(G). Aouchiche and Hansen [2] introduced the signlessLaplacian for the distance matrix of G as DQ(G) = Tr(G) + D(G), where Tr(G) =diag(Tr(v1),Tr(v2), . . . ,Tr(vn)) is the diagonal matrix of the vertex transmissions in G.Also, the eigenvalues of DQ(G) can be arranged as ∂

Q1 (G) ≥ ∂

Q2 (G) ≥ · · · ≥ ∂

Qn (G),

and m(∂Qi (G)) denotes the multiplicity of ∂

Qi (G). Further, the distance signless Laplacian

spectrum of G, denoted by SpecQ(G), could be similarly defined.The graphs with few distinct adjacency eigenvalues always possess nice combinatorial

properties and forman interesting class to study.Manymathematicians have been attractedto this topic, and they obtain many beautiful results, see [3–9]. However, it seems to be

CONTACT Qiongxiang Huang huangqx@xju.edu.cn© 2017 Informa UK Limited, trading as Taylor & Francis Group

https://doi.org/10.1080/03081087.2017.1391166

LINEAR AND MULTILINEAR ALGEBRA2018, VOL. 66, NO. 11, 22182231

J Algebr Comb (2018) 47:585–601https://doi.org/10.1007/s10801-017-0787-x

Integral Cayley graphs over dihedral groups

Lu Lu1 · Qiongxiang Huang1 · Xueyi Huang1

Received: 27 November 2016 / Accepted: 20 August 2017 / Published online: 31 August 2017© Springer Science+Business Media, LLC 2017

Abstract In this paper, we give a necessary and sufficient condition for the integralityof Cayley graphs over the dihedral group Dn = 〈a, b | an = b2 = 1, bab = a−1〉.Moreover,we alsoobtain some simple sufficient conditions for the integrality ofCayleygraphs over Dn in terms of the Boolean algebra of 〈a〉, from which we find infiniteclasses of integral Cayley graphs over Dn . In particular, we completely determine allintegral Cayley graphs over the dihedral group Dp for a prime p.

Keywords Cayley graph · Integral graph · Dihedral group · Character

Mathematics Subject Classification 05C50

1 Introduction

A graph X is said to be integral if all eigenvalues of the adjacency matrix of X are inte-gers. The property was first defined by Harary and Schwenk [12], who suggested theproblem of classifying integral graphs. This problem initiated a significant investiga-tion among algebraic graph theorists, trying to construct and classify integral graphs.Although this problem is easy to state, it turns out to be extremely hard. It has beenattacked by many mathematicians during the past 40 years, and it is still wide open.

Supported by the National Natural Science Foundation of China (Grant Nos. 11671344, 11531011).

B Qiongxiang Huanghuangqx@xju.edu.cn

1 College of Mathematics and Systems Science, Xinjiang University, Ürümqi 830046, Xinjiang,People’s Republic of China

123

Graphs and Combinatorics (2018) 34:395–414https://doi.org/10.1007/s00373-018-1880-1

Graphs with at Most Three Distance EigenvaluesDifferent from −1 and −2

Xueyi Huang1 · Qiongxiang Huang1 · Lu Lu1

Received: 19 September 2017 / Revised: 6 February 2018 / Published online: 6 March 2018© Springer Japan KK, part of Springer Nature 2018

Abstract Let G be a connected graph on n vertices, and let D(G) be the distancematrix of G. Let ∂1(G) ≥ ∂2(G) ≥ · · · ≥ ∂n(G) denote the eigenvalues of D(G). Inthis paper, we characterize all connected graphs with ∂3(G) ≤ −1 and ∂n−1(G) ≥ −2.In the course of this investigation, we determine all connected graphs with at mostthree distance eigenvalues different from −1 and −2.

Keywords Distance matrix · The third largest distance eigenvalue · The second leastdistance eigenvalue

Mathematics Subject Classification 05C50

1 Introduction

Let G be a connected simple graph with vertex set V (G) = {v1, v2, . . . , vn}. Denoteby dG(vi , v j ) the length of the shortest path connecting vi and v j in G. The distancebetween v ∈ V (G) and H , a connected subgraph of G, is defined to be d(v, H) =min{dG(v,w) | w ∈ V (H)}. Furthermore, we define the diameter and distancematrixof G as d(G) = max{dG(vi , v j ) | vi , v j ∈ V (G)} and D(G) = [dG(vi , v j )]n×n ,respectively. Then the characteristic polynomial ΦG(x) = det(x I − D(G)) of D(G)

is also called the distance polynomial (D-polynomial for short) of G.Since D(G) is a real and symmetric, its eigenvalues can be conveniently denoted

and arranged as ∂1(G) ≥ ∂2(G) ≥ · · · ≥ ∂n(G). These eigenvalues are also called

B Qiongxiang Huanghuangqx@xju.edu.cn

1 College of Mathematics and Systems Science, Xinjiang University, Ürümqi 830046, Xinjiang,People’s Republic of China

123

Electronic Journal of Linear Algebra, ISSN 1081-3810A publication of the International Linear Algebra SocietyVolume 32, pp. 531-538, December 2017.http://repository.uwyo.edu/ela

ON THE SECOND LEAST DISTANCE EIGENVALUE OF A GRAPH∗

XUEYI HUANG† , QIONGXIANG HUANG† , AND LU LU†

Abstract. Let G be a connected graph on n vertices, and let D(G) be the distance matrix of G. Let ∂1(G) ≥ ∂2(G) ≥· · · ≥ ∂n(G) denote the eigenvalues of D(G). In this paper, the connected graphs with ∂n−1(G) at least the smallest root of

x3 − 3x2 − 11x− 6 = 0 are determined. Additionally, some non-isomorphic distance cospectral graphs are given.

Key words. Distance matrix, Second least distance eigenvalue, Distance cospectral graph.

AMS subject classifications. 05C50.

1. Introduction. Let G be a connected simple graph with vertex set V (G) = {v1, v2, . . . , vn}. Denoted

by d(vi, vj) the length of the shortest path connecting vi and vj in G. Let H be a connected subgraph of

G and v ∈ V (G). The distance between v and H is defined to be d(v,H) = min{d(v, w) | w ∈ V (H)}.Also, the diameter and distance matrix of G are defined as d(G) = max{d(vi, vj) | vi, vj ∈ V (G)} and

D(G) = [d(vi, vj)]n×n, respectively. The characteristic polynomial ΦG(x) = det(xI −D(G)) of D(G) is also

called the distance polynomial of G.

Since D(G) is a real and symmetric, its eigenvalues can be listed as ∂1(G) ≥ ∂2(G) ≥ · · · ≥ ∂n(G). These

eigenvalues are also called the distance eigenvalues of G. The distance spectrum of G, denoted by SpecD(G),

is the multiset of distance eigenvalues of G. Two connected graphs are said to be distance cospectral if they

share the same distance spectrum, and the graph G is called determined by its distance spectrum if any

connected graph distance cospectral with G must be isomorphic to it.

Let NG(v) denote the neighborhood of v ∈ V (G), G[X] the induced subgraph of G on X ⊆ V (G), and

DG(X) the principle submatrix of D(G) corresponding to G[X]. Also, we denote by Kn and Pn the complete

graph and path on n vertices, respectively.

For a connected graph G whose vertices are labeled as v1, v2, . . . , vn, and a sequence of graphs H1, H2,

. . . , Hn, the corresponding generalized lexicographic product G[H1, . . . ,Hn] is defined as the graph obtained

from G by replacing vi with the graph Hi for 1 ≤ i ≤ n, and connecting all edges between Hi and Hj if

vi is adjacent to vj for 1 ≤ i 6= j ≤ n. For example, Figure 1 illustrates the graph P4[Ka1 ,Ka2 ,Ka3 ,Ka4 ],

where Ai denotes the vertex subset of P4[Ka1 ,Ka2 ,Ka3 ,Ka4 ] corresponding to Kai for 1 ≤ i ≤ 4 and the

line segments represent connecting all edges between Ai and Ai+1 for 1 ≤ i ≤ 3.

Connected graphs whose distance eigenvalues satisfy special conditions and the study of whether such

graphs are determined by their distance spectra have received some attention recently. Lin et al. [4] (see

also Yu [9]) proved that ∂n(G) = −2 if and only if G is a complete multipartite graph, and conjectured

that complete multipartite graphs are determined by their distance spectra. Recently, Jin and Zhang [1]

∗Received by the editors on August 21, 2017. Accepted for publication on November 24, 2017. Handling Editor: Bryan

L. Shader. This work was supported by the National Natural Science Foundation of China (grants no. 11671344 and no.

11701492).†College of Mathematics and Systems Science, Xinjiang University, Urumqi, P.R. China (huangqx@xju.edu.cn,

huangxymath@gmail.com, lulu549588@hotmail.com).

AlgebraColloquiumc⃝ 2017 AMSS CAS

& SUZHOU UNIV

Algebra Colloquium 24 : 4 (2017) 541–550

DOI: 10.1142/S1005386717000359

Automorphism Groups of a Class of Cubic

Cayley Graphs on Symmetric Groups∗

Xueyi Huang Qiongxiang Huang† Lu LuCollege of Mathematics and Systems Science, Xinjiang University

Urumqi, Xinjiang 830046, China

E-mail: huangqx@xju.edu.cn huangqxmath@163.com

Received 1 November 2016Revised 18 June 2017

Communicated by Genghua Fan

Abstract. Let Sn denote the symmetric group of degree n with n ≥ 3, S ={cn =

(1 2 · · · n), c−1n , (1 2)

}and Γn = Cay(Sn, S) be the Cayley graph on Sn with respect

to S. In this paper, we show that Γn (n ≥ 13) is a normal Cayley graph, and that thefull automorphism group of Γn is equal to Aut(Γn) = R(Sn)o ⟨Inn(ϕ)⟩ ∼= Sn × Z2, whereR(Sn) is the right regular representation of Sn, ϕ = (1 2)(3 n)(4 n−1)(5 n−2) · · · (∈ Sn),and Inn(ϕ) is the inner isomorphism of Sn induced by ϕ.

2010 Mathematics Subject Classification: 05C25

Keywords: Cayley graph, normal, automorphism group

1 Introduction

Let G be a finite group, and S a subset of G with e ∈ S (e is the identity element ofG) and S = S−1. The Cayley graph on G with respect to S, denoted by Cay(G,S),is defined to be the undirected graph with vertex set G and with an edge connectingg, h ∈ G if hg−1 ∈ S. Denote by Aut(Cay(G,S)) and Aut(G) the automorphismgroups of Cay(G,S) andG, respectively. The right regular representation (resp., leftregular representation) of the group G is defined as R(G) = {rg : x 7→ xg (∀ x ∈ G) |g ∈ G} (resp., L(G) = {lg : x 7→ g−1x (∀ x ∈ G) | g ∈ G}). Clearly, R(G)is a subgroup of Aut(Cay(G,S)) and so every Cayley graph is vertex-transitive.Furthermore, the group Aut(G,S) = {σ ∈ Aut(G) | Sσ = S} is a subgroup ofAut(Cay(G,S))e, the stabilizer of the identity vertex e in Aut(Cay(G,S)), hencealso a subgroup of Aut(Cay(G,S)). The Cayley graph Cay(G,S) is said to be

∗This work is supported by the National Natural Science Foundation of China (Grant Nos.

11671344 and 11531011).†Corresponding author.

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Electronic Journal of Linear Algebra, ISSN 1081-3810A publication of the International Linear Algebra SocietyVolume 32, pp. 365-379, September 2017.http://repository.uwyo.edu/ela

ON THE CONSTRUCTION OF Q-CONTROLLABLE GRAPHS∗

ZHENZHEN LOU† , QIONGXIANG HUANG† , AND XUEYI HUANG†

Abstract. A connected graph is called Q-controllable if its signless Laplacian eigenvalues are mutually distinct and main.

Two graphs G and H are said to be Q-cospectral if they share the same signless Laplacian spectrum. In this paper, infinite

families of Q-controllable graphs are constructed, by using the operator of rooted product introduced by Godsil and McKay. In

the process, infinitely many non-isomorphic Q-cospectral graphs are also constructed, especially, including those graphs whose

signless Laplacian eigenvalues are mutually distinct.

Key words. Q-Spectrum, Q-Controllable graph, Q-Cospectral graph.

AMS subject classifications. 05C50.

1. Introduction. All graphs considered here are simple and undirected. For a graph G = (V (G), E(G))

of order n with vertex set V (G) = {1, 2, . . . , n}, we denote by A(G) and D(G) = diag(d1, d2, . . . , dn) the

adjacency matrix and diagonal degree matrix of G, respectively, where di is the degree of the vertex i. Then

the matrix Q(G) = D(G) + A(G) is called the signless Laplacian matrix (Q-matrix for short) of the graph

G. Since Q(G) is positive semidefinite, all its eigenvalues are nonnegative. These eigenvalues are called the

signless Laplacian eigenvalues (Q-eigenvalues for short) of G. Let ξ1 > ξ2 > · · · > ξs ≥ 0 be all the distinct

Q-eigenvalues of G with multiplicities m1,m2, . . . ,ms (∑s

i=1mi = n), respectively. The signless Laplacian

spectrum (Q-spectrum for short) of G is defined to be SpecQ(G) = {ξm11 , ξm2

2 , . . . , ξmss }. Two graphs G and

H are called Q-cospectral if SpecQ(G) = SpecQ(H), and a graph G is said to be determined by its Q-spectrum

(DQS for short) if G ∼= H whenever SpecQ(G) = SpecQ(H) for any graph H.

Given a graph G of order n and a graph H with root vertex u, the rooted product graph G ◦H is defined

as the graph obtained from G and H by taking one copy of G and n copies of H and identifying the vertex

vi of G with the vertex u in the i-th copy of H for every 1 ≤ i ≤ n (Godsil and McKay [5]). Let Ps be the

path of order s. If we take H = Ps (s ≥ 1), and the root vertex u = u1 one of pendant vertices of H, then

the rooted product graph G ◦ Ps is shown in Fig. 1 (see Section 3).

A Q-eigenvalue of G is called a main Q-eigenvalue if it has an eigenvector x such that jTx 6= 0 (j is the

n× 1 all-ones vector), and a non-main Q-eigenvalue otherwise. Connected graphs whose Q-eigenvalues are

mutually distinct and main are called Q-controllable graphs. Throughout the paper, we denote by GQ (resp.,

GQn ) the set of connected graphs (resp., with n vertices) whose eigenvalues are mutually distinct, and GQ∗

(resp., GQ∗

n ) the set of Q-controllable graphs (resp., with n vertices).

For a graph G on n vertices with adjacency matrix A and diagonal degree matrix D, a universal adjacency

matrix associated with G is defined to be U = γAA+γDD+γII+γJJ , where I denotes the identity matrix,

J denotes the all-ones matrix, and γA 6= 0, γD, γI and γJ are constants [6]. Note that U = Q(G) if we take

γA = γD = 1 and γI = γJ = 0. The name “Q-controllable graph” arised from the concept of U -controllable

∗Received by the editors on May 8, 2016. Accepted for publication on February 28, 2017. Handling Editor: Sebastian M.

Cioaba.†College of Mathematics and Systems Science, Xinjiang University, Urumqi, P.R. China (xjdxlzz@163.com,

huangqx@xju.edu.cn, huangxymath@gmail.com).

Applied Mathematics and Computation 314 (2017) 58–64

Contents lists available at ScienceDirect

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journal homepage: www.elsevier.com/locate/amc

Automorphism group of the complete alternating group

graph

�

Xueyi Huang, Qiongxiang Huang

∗

College of Mathematics and Systems Science, Xinjiang University, Urumqi, Xinjiang 830046, PR China

a r t i c l e i n f o

MSC:

05C25

Keywords:

Complete alternating group graph

Automorphism group

Normal Cayley graph

a b s t r a c t

Let S n and A n denote the symmetric group and alternating group of degree n with n ≥ 3,

respectively. Let S be the set of all 3-cycles in S n . The complete alternating group graph ,

denoted by CAG n , is defined as the Cayley graph Cay( A n , S ) on A n with respect to S .

In this paper, we show that CAG n ( n ≥ 4) is not a normal Cayley graph. Furthermore,

the automorphism group of CAG n for n ≥ 5 is obtained, which equals to Aut (CAG n ) =

(R (A n ) � Inn (S n )) � Z 2 ∼=

(A n � S n ) � Z 2 , where R ( A n ) is the right regular representation

of A n , Inn( S n ) is the inner automorphism group of S n , and Z 2 = 〈 h 〉 , where h is the map

α �→ α−1 ( ∀ α ∈ A n ). © 2017 Elsevier Inc. All rights reserved.

1. Introduction

Let X = (V, E) be a simple undirected graph. An automorphism of X is a permutation on its vertex set V that preserves

adjacency relations. The automorphism group of X , denoted by Aut(X), is the set of all automorphisms of X .

For a finite group �, and a subset T of � such that e �∈ T ( e is the identity element of �) and T = T −1 , the Cayley graph

Cay( �, T ) on � with respect to T is defined as the undirected graph with vertex set � and edge set {( γ , t γ ) | γ ∈ �, t ∈ T }.

The right regular representation R (�) = { r γ : x �→ xγ (∀ x ∈ �) | γ ∈ �} , i.e., the action of � on itself by right multiplication,

is a subgroup of the automorphism group Aut(Cay( �, T )) of the Cayley graph Cay( �, T ). Hence, every Cayley graph is vertex-

transitive. Furthermore, the group Aut (�, T ) = { σ ∈ Aut (�) | T σ = T } is a subgroup of Aut(Cay( �, T )) e , the stabilizer of the

identity vertex e in Aut(Cay( �, T )), and so is also a subgroup of Aut(Cay( �, T )). The Cayley graph Cay( �, T ) is said to be

normal if R ( �) is a normal subgroup of Aut(Cay( �, T )). By Godsil [17] , N Aut ( Cay (�,T )) (R (�)) = R (�) � Aut (�, T ) . Thus, Cay( �,

T ) is normal if and only if Aut ( Cay (�, T )) = R (�) � Aut (�, T ) .

A basic problem in algebraic graph theory is to determine the (full) automorphism groups of Cayley graphs. As the (full)

automorphism group of a normal Cayley graph Cay( �, T ) is always equal to Aut ( Cay (�, T )) = R (�) � Aut (�, T ) , to deter-

mine the normality of Cayley graphs is an important problem in the literature. The whole information about the normality

of Cayley graphs on the cyclic groups of prime order and the groups of order twice a prime were gained by Alspach [1] and

Du et al. [5] , respectively. Wang et al. [24] obtained all disconnected normal Cayley graphs. Fang et al. [8] proved that the

vast majority of connected cubic Cayley graphs on non-abelian simple groups are normal. Baik et al. [3,4] listed all con-

nected non-normal Cayley graphs on abelian groups with valency less than 6 and Feng et al. [10] proved that all connected

� Supported by the National Natural Science Foundation of China (Grant nos. 11671344 , 11531011 ). ∗ Corresponding author.

E-mail addresses: huangxymath@gmail.com (X. Huang), huangqxmath@163.com , huangqx@xju.edu.cn (Q. Huang).

http://dx.doi.org/10.1016/j.amc.2017.07.009

0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

Linear Algebra and its Applications 530 (2017) 485–499

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On graphs with distance Laplacian spectral radius

of multiplicity n − 3 ✩

Lu Lu, Qiongxiang Huang ∗, Xueyi HuangCollege of Mathematics and Systems Science, Xinjiang University, Urumqi, Xinjiang 830046, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 March 2017Accepted 28 May 2017Available online 1 June 2017Submitted by D. Stevanovic

MSC:05C5005C1215A18

Keywords:Distance Laplacian matrixLaplacian matrixLargest eigenvalueCharacterised by distance Laplacian spectrum

Let ∂L1 ≥ ∂L

2 ≥ · · · ≥ ∂Ln be the distance Laplacian eigenvalues

of a connected graph G and m(∂Li ) the multiplicity of ∂L

i . It is well known that the graphs with m(∂L

1 ) = n − 1are complete graphs. Recently, the graphs with m(∂L

1 ) =n − 2 have been characterised by Celso et al. In this paper, we completely determine the graphs with m(∂L

1 ) = n − 3.© 2017 Elsevier Inc. All rights reserved.

1. Introduction

In this paper we only consider simple connected graphs. Let G = (V, E) be a connected graph with vertex set V = {v1, v2, . . . , vn} and edge set E = {e1, e2, . . . , em}. The

✩ Supported by the National Natural Science Foundation of China (Grant Nos. 11671344, 11531011).* Corresponding author.

E-mail address: huangqx@xju.edu.cn (Q. Huang).

http://dx.doi.org/10.1016/j.laa.2017.05.0440024-3795/© 2017 Elsevier Inc. All rights reserved.

Acta Mathematica Sinica, English Series

Jul., 2017, Vol. 33, No. 7, pp. 996–1010

Published online: February 27, 2017

DOI: 10.1007/s10114-017-6241-0

Http://www.ActaMath.com

Acta Mathematica Sinica, English Series© Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2017

Enumeration of Cubic Cayley Graphs on Dihedral Groups

Xue Yi HUANG Qiong Xiang HUANG1) Lu LUCollege of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, P. R. China

E-mail : huangxymath@gmail.com huangqx@xju.edu.cn lulu 549588@hotmail.com

Abstract Let p be an odd prime, and D2p = 〈a, b | ap = b2 = 1, bab = a−1〉 the dihedral group of

order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by

means of spectral method. By the way, we show that two cubic Cayley graphs on D2p are isomorphic if

and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley

graphs on D2p by using Gauss’ celebrated law of quadratic reciprocity.

Keywords Cayley graph, dihedral group, cospectral, isomorphic classes, quadratic reciprocity

MR(2010) Subject Classification 05C25, 05C50

1 Introduction

Let G be a finite group, and let S be a subset of G such that 1 �∈ S and S is symmetric, thatis, S−1 = {s−1 | s ∈ S} = S. The Cayley graph on G with respect to S, denoted by X(G,S), isthe undirected graph with vertex set G and with an edge {g, h} connecting g and h if hg−1 ∈ S,or equivalently gh−1 ∈ S. In particular, if G is a cyclic group, then the Cayley graph X(G,S)is called a circulant graph.

Let X(G,S) be the Cayley graph on G with respect to S. Suppose that σ ∈ Aut(G), whereAut(G) is the automorphism group of G. Let T = σ(S). Then it is easily shown that σ inducesan isomorphism fromX(G,S) toX(G, T ). Such an isomorphism is called a Cayley isomorphism.However, it is possible for two Cayley graphs X(G,S) and X(G, T ) to be isomorphic but noCayley isomorphisms mapping S to T . The Cayley graph X(G,S) is called a CI-graph of Gif, for any Cayley graph X(G, T ), whenever X(G,S) ∼= X(G, T ) we have σ(S) = T for someσ ∈ Aut(G). A group G is called a CI-group if all Cayley graphs on G are CI-graphs. A long-standing open question about Cayley graphs is as follows: which Cayley graphs for a groupG are CI-graphs? This question stems from a conjecture proposed by Adam [2]: all circulantgraphs are CI-graphs of the corresponding cyclic groups. This conjecture was disproved byElspas and Turner [9], and however, the conjecture stimulated the investigation of CI-graphsand CI-groups [3, 7, 8, 11–14, 16, 18, 19]. Another motivation for investigating CI-graphs isto determine the isomorphic classes of Cayley graphs. By the definition, if X(G,S) is a CI-graph, then to decide whether or not X(G,S) is isomorphic to X(G, T ), we only need to decidewhether or not there exists an automorphism σ ∈ Aut(G) such that σ(S) = T . The isomorphic

Received April 27, 2016, revised November 2, 2016, accepted December 12, 2016

Supported by National Natural Science Foundation of China (Grant Nos. 11671344 and 11531011)

1) Corresponding author

Math. Nachr. 290, No. 5–6, 955–964 (2017) / DOI 10.1002/mana.201500313

Q-integral unicyclic, bicyclic and tricyclic graphs

Jing Zhang∗, Qiongxiang Huang∗∗, Caixia Song∗∗∗, and Xueyi Huang†

College of Mathematics and Systems Science, Xinjiang University, Urumqi, Xinjiang 830046, P. R. China

Received 20 August 2015, revised 29 May 2016, accepted 23 September 2016Published online 20 January 2017

Key words Q-integral, k-cyclic graph, signless Laplacian eigenvalueMSC (2010) 05C50

A graph is called Q-integral if its signless Laplacian spectrum consists of integers. In this paper, we characterizea class of k-cyclic graphs whose second smallest signless Laplacian eigenvalue is less than one. Using this resultwe determine all the Q-integral unicyclic, bicyclic and tricyclic graphs.

C© 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Let G = (V (G), E(G)) be an undirected simple graph of order n with adjacency matrix A(G) and diagonaldegree matrix D(G) = diag(d1, . . . , dn). The Laplacian matrix and signless Laplacian matrix of G are definedas L(G) = D(G) − A(G) and Q(G) = D(G) + A(G), respectively.

It is easy to verify that both L(G) and Q(G) are positive semidefinite, and so their eigenvalues are conven-tionally denoted and arranged as μ1(G) ≥ μ2(G) ≥ · · · ≥ μn−1(G) ≥ μn(G) = 0 and q1(G) ≥ q2(G) ≥ · · · ≥qn−1(G) ≥ qn(G) ≥ 0, respectively.

These eigenvalues of Q(G) are also called the signless Laplacian eigenvalues (in short, Q-eigenvalues) ofG. All the Q-eigenvalues together with their multiplicities are called the signless Laplacian spectrum (in short,Q-spectrum) of G denoted by SpecQ(G). A graph is called signless Laplacian integral (in short, Q-integral) if itsQ-spectrum consists of integers. The notions L-eigenvalue, L-spectrum and L-integral can be similarly definedif we consider the Laplacian matrix L(G).

As usual, we denote by G ∪ H the disjoint union of graphs G and H , kG the disjoint union of k copies of Gand G ′ = G + e the graph obtained from G by adding a new edge e. Let dG(v) denotes the degree of v ∈ V (G).Especially, �(G) and δ(G) denote the maximum degree and minimum degree of vertices of G, respectively. IfdG(v) = 1, then we call v a pendant vertex of G. Further, an (r, s)-semiregular bipartite graph is a bipartite graphwhose each vertex in the first (resp. second) colour class has degree r (resp. s).

A connected graph with n vertices and m edges is called a k-cyclic graph if k = m − n + 1. A k-cyclic graphis said to be a k-cyclic base graph if it contains no pendant vertices. In particular, the notions unicyclic, bicyclicand tricyclic (base) graph are respectively defined as the k-cyclic (base) graph with k = 1, 2 and 3.

The research of integral graphs started in 1974 [15]. It has been discovered recently [5] that integral graphs canplay a role in the so-called perfect state transfer in quantum spin networks. Indeed, the number of integral graphsis not only infinite, but one can find them in all classes of graphs and among graphs of all orders. However, theproblem to determine integral graphs has far from been resolved. Up to now, just a few classes of integral graphshave been characterized. For a survey and references about integral graphs see [1]–[4], [11], [12], [17]–[20],[22], [24]–[33]. With regard to Q-spectrum, all the connected Q-integral graphs up to 10 vertices [26], all theQ-integral graphs with edge-degrees at most four [24] and all the Q-integral complete split graphs, multiplecomplete split-like graphs, extended complete split-like graphs and multiple extended split-like graphs [12] are

∗ e-mail: zhangjxju@163.com∗∗ Corresponding author: e-mail: huangqx@xju.edu.cn and huangqxmath@163.com∗∗∗ e-mail: songcxmath@163.com† e-mail: huangxymath@gmail.com

C© 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

J Algebr Comb (2017) 45:629–647DOI 10.1007/s10801-016-0718-2

The graphs with exactly two distance eigenvaluesdifferent from −1 and −3

Lu Lu1 · Qiongxiang Huang1 · Xueyi Huang1

Received: 31 August 2015 / Accepted: 28 September 2016 / Published online: 13 October 2016© Springer Science+Business Media New York 2016

Abstract In this paper, we completely characterize the graphs with third largest dis-tance eigenvalue at most−1 and smallest distance eigenvalue at least−3. In particular,we determine all graphs whose distance matrices have exactly two eigenvalues (count-ing multiplicity) different from −1 and −3. It turns out that such graphs consist ofthree infinite classes, and all of them are determined by their distance spectra. We alsoshow that the friendship graph is determined by its distance spectrum.

Keywords Distance eigenvalue · Distance equitable partition · Friendship graph ·Distance spectral characterization

Mathematics Subject Classification 05C50

1 Introduction

Let G be a simple connected graph with vertex set V (G) = {v1, v2, . . . , vn}, edge setE(G) and adjacency matrix A = A(G). Denote by d(vi , v j ) the distance (i.e., thelength of a shortest path) between the vertices vi and v j ofG. Then, the diameter d(G)

and distance matrix D(G) are defined as d(G) = max{d(vi , v j ) | vi , v j ∈ V (G)}and D(G) = (d(vi , v j ))n×n , respectively.

Supported by the National Natural Science Foundation of China (11671344, 11261059, 11531011).

B Qiongxiang Huanghuangqx@xju.edu.cn

1 College of Mathematics and Systems Science, Xinjiang University, Ürümqi 830046, Xinjiang,People’s Republic of China

123

Discrete Mathematics 340 (2017) 607–616

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Construction of graphs with distinct eigenvalues✩

Zhenzhen Lou, Qiongxiang Huang *, Xueyi HuangCollege of Mathematics and Systems Science, Xinjiang University, Urumqi, Xinjiang 830046, PR China

a r t i c l e i n f o

Article history:Received 29 February 2016Received in revised form 22 September2016Accepted 29 November 2016Available online 3 January 2017

Keywords:Main eigenvalueControllable graphCospectral graph

a b s t r a c t

Let G (resp. Gn) be the set of connected graphs (resp. with n vertices) whose eigenvaluesare mutually distinct, and G∗ (resp. G∗

n ) the set of connected graphs (resp. with n vertices)whose eigenvalues are mutually distinct and main. Two graphs G and H are said to becospectral if they share the same adjacency spectrum. In this paper, we give a newmethodto construct infinite families of graphs in G and G∗. Concretely, given a graph G in Gn orG∗n , the infinite families of G or G∗ are constructed from G, and furthermore the spectra of

such graphs are also characterized by the spectrum of G. By the way, we use this method toconstruct some infinite families of non-isomorphic cospectral graphs, especially, includingthe graphs in G and G∗.

© 2016 Elsevier B.V. All rights reserved.

1. Introduction

All graphs considered here are simple and undirected. For a graph G = (V (G), E(G)) of order n with adjacency matrixA = A(G), we denote by λ1, λ2, . . . , λd (d ≤ n) all the distinct eigenvalues of A with multiplicities m1,m2, . . . ,md(∑d

i=1mi = n), respectively. These eigenvalues are also called the eigenvalues of G. All the eigenvalues together with theirmultiplicities are called the adjacency spectrum of G denoted by SpecA(G) =

{λm11 , λ

m22 , . . . , λ

mdd

}. The adjacency spectrum

of G together with that of its complement will be referred to as the generalized spectrum of G denoted by SpecG(G).Two graphs G and H are called cospectral if SpecA(G) = SpecA(H), and a graph G is said to be determined by its adjacency

spectrum (DAS for short) if G ∼= H whenever SpecA(G) = SpecA(H) for any graph H . The notion determined by its generalizedspectrum (DGS for short) can be similarly defined.

Given a graph G of order n and a graphH with root vertex u, the rooted product graph G◦H is defined as the graph obtainedfrom G and H by taking one copy of G and n copies of H and identifying the vertex vi of Gwith the vertex u in the ith copy ofH for every 1 ≤ i ≤ n (see [9,22]). Let Ps be the path of order s. If we take H = Ps (s ≥ 1), and the root vertex u = u1 one ofpendant vertices of H , then the rooted product graph G ◦ Ps is shown in Fig. 1.

An eigenvalue of G is called amain eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero, and anon-main eigenvalue otherwise. Connected graphs whose eigenvalues are mutually distinct and main are called controllablegraphs. In some previous work [5,6], connectedness is a prerequisite for a controllable graph, but in general this conditioncan be avoided [29]. Throughout the paper, we denote by G (resp. Gn) the set of connected graphs (resp. with n vertices)whose eigenvalues are mutually distinct, and G∗ (resp. G∗

n ) the set of controllable graphs (resp. with n vertices).

✩ This work is supported by NSFC (Nos. 11671344, 11261059, 11531011 and 11501486).

* Corresponding author.E-mail address: huangqx@xju.edu.cn (Q. Huang).

http://dx.doi.org/10.1016/j.disc.2016.11.0330012-365X/© 2016 Elsevier B.V. All rights reserved.

Linear Algebra and its Applications 512 (2017) 219–233

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On regular graphs with four distinct eigenvalues ✩

Xueyi Huang, Qiongxiang Huang ∗

College of Mathematics and Systems Science, Xinjiang University, Urumqi, Xinjiang 830046, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 November 2015Accepted 30 September 2016Available online 4 October 2016Submitted by R. Brualdi

MSC:05C50

Keywords:Regular graphsEigenvaluesDS

Let G(4, 2) be the set of connected regular graphs with four distinct eigenvalues in which exactly two eigenvalues are simple, G(4, 2, −1) (resp. G(4, 2, 0)) the set of graphs belonging to G(4, 2) with −1 (resp. 0) as an eigenvalue, and G(4, ≥ −1) the set of connected regular graphs with four distinct eigenvalues and second least eigenvalue not less than −1. In this paper, we prove the non-existence of connected graphs having four distinct eigenvalues in which at least three eigenvalues are simple, and determine all the graphs in G(4, 2, −1). As a by-product of this work, we characterize all the graphs belonging to G(4, ≥ −1) and G(4, 2, 0), respectively, and show that all these graphs are determined by their spectra.

© 2016 Published by Elsevier Inc.

1. Introduction

Let G = (V (G), E(G)) be a simple undirected graph on n vertices with adjacency matrix A = A(G). Denote by λ1, λ2, . . . , λt all the distinct eigenvalues of A with mul-tiplicities m1, m2, . . . , mt (

∑ti=1 mi = n), respectively. These eigenvalues are also called

the eigenvalues of G. All the eigenvalues together with their multiplicities are called the

✩ This work is supported by NSFC (Grant Nos. 11671344, 11261059 and 11531011).* Corresponding author.

E-mail addresses: huangxymath@gmail.com (X. Huang), huangqx@xju.edu.cn (Q. Huang).

http://dx.doi.org/10.1016/j.laa.2016.09.0430024-3795/© 2016 Published by Elsevier Inc.

Linear Algebra and its Applications 486 (2015) 204–218

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Construction of graphs with exactly k main

eigenvalues ✩

Xueyi Huang, Qiongxiang Huang ∗, Lu LuCollege of Mathematics and Systems Science, Xinjiang University, Urumqi, Xinjiang 830046, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 July 2015Accepted 17 August 2015Available online 27 August 2015Submitted by R. Brualdi

MSC:05C50

Keywords:Equitable partitionDivisorDirected multigraphMain eigenvalue

Given a simple graph G, the vertex partition Π: V (G) = V1 ∪V2 ∪ · · · ∪ Vr is said to be an equitable partition if, for any u ∈ Vi, |Vj ∩NG(u)| = bij is a constant whenever 1 ≤ i, j ≤ r. An equitable partition Π leads to a divisor G/Π of G, which is the directed multigraph with vertices V1, V2, . . . , Vr and bijarcs from Vi to Vj . Conversely, a directed multigraph may not be a divisor of some simple graph. In this paper we give a necessary and sufficient condition for a directed multigraph to be the divisor of some simple graph. By the way, we give a method to construct many classes of connected graphs with exactly k main eigenvalues for any positive integer k.

© 2015 Published by Elsevier Inc.

1. Introduction

Let G be a simple graph with vertex set V (G), edge set E(G) and adjacency matrix A = A(G) with eigenvalues λ1, λ2, . . . , λn. We call λ1, λ2, . . . , λn the eigenvalues of G. The characteristic polynomial of G, denoted by PG(x), can be expressed as PG(x) =det(xI −A(G)) =

∏ni=1(x − λi).

✩ This work is supported by National Natural Science Foundation of China (No. 11261059).* Corresponding author.

E-mail address: huangqx@xju.edu.cn (Q. Huang).

http://dx.doi.org/10.1016/j.laa.2015.08.0130024-3795/© 2015 Published by Elsevier Inc.

Linear and Multilinear Algebra, 2015Vol. 63, No. 7, 1356–1371, http://dx.doi.org/10.1080/03081087.2014.936436

On the Laplacian integral tricyclic graphs

Xueyi Huang, Qiongxiang Huang∗ and Fei Wen

College of Mathematics and Systems Science, Xinjiang University, Urumqi, P.R. China

Communicated by S. Pati

(Received 10 December 2013; accepted 31 May 2014)

A graph is called Laplacian integral if all its Laplacian eigenvalues are integers.In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of agraph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraicconnectivity is less than one. Using these results, we determine all the Laplacianintegral tricyclic graphs. Furthermore, we show that all the Laplacian integraltricyclic graphs are determined by their Laplacian spectra.

Keywords: Laplacian integral graph; algebraic connectivity; tricyclic graph

AMS Subject Classification: 05C50

1. Introduction

Let G = (V (G), E(G)) be an undirected simple graph of order n. The Laplacian matrixof G is defined as L(G) = D(G) − A(G), where D(G) = diag(d1, . . . , dn) and A(G)

denote the diagonal degree matrix and adjacency matrix of G, respectively.It is easy to verify that L(G) is positive semidefinite, and so its eigenvalues are con-

ventionally denoted and arranged as

λ1(G) ≥ λ2(G) ≥ · · · ≥ λn−1(G) ≥ λn(G) = 0.

These eigenvalues of L(G) are also called the Laplacian eigenvalues (in short,L-eigenvalues) of G. It is well known that λn−1(G) > 0 if and only if G is connectedand hence is called the algebraic connectivity of G (see, for instance, [1]).

All the L-eigenvalues togetherwith theirmultiplicities are called theLaplacian spectrum(in short, L-spectrum) of G denoted by SpecL(G). A graph is called Laplacian integral(in short, L-integral) if its L-spectrum consists of integers, and a graph G is said to bedetermined by its L-spectrum (in short, DLS) if G ∼= H whenever SpecL(G) = SpecL(H)

for any graph H .As usual, we denote by G ∪ H the disjoint union of graphs G and H, kG the disjoint

union of k copies of G,G ′ = G+e the graph obtained from G by adding a new edge e, andG − v the graph obtained from G by deleting the vertex v ∈ V (G) along with the edgesadjacent to v.

∗Corresponding author. Email: huangqx@xju.edu.cn

© 2014 Taylor & Francis

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