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, we introduce three types of congruent transformations for graphs that keep the positive inertia index and negative inertia index. By using these congruent transformations, we determine all graphs with exactly two positive eigenvalues and one zero eigenvalue.
A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected bicyclic graphs with exactly two main eigenvalues are determined.
In 1977 Smith characterized graphs with exactly one positive eigenvalue. Since then, many particular results related to graphs with exactly two positive eigenvalues have emerged. In this paper we conclude this investigation by giving a full characterization of these graphs.
The signless Laplacian matrix of a graph $G$ is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called $Q$-eigenvalues of $G$. A $Q$-eigenvalue of a graph $G$ is called a $Q$-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Chen and Huang [L. Chen, Q.X. Huang, Trees, unicyclic graphs and bicyclic graphs with exactly two $Q$-main eigenvalues, submitted for publication] characterized all trees, unicylic graphs and bicyclic graphs with exactly two main $Q$-eigenvalues, respectively. As a continuance of it, in…
In this note, we consider connected graphs with exactly two main eigenvalues. We will give several constructions for them, and as a consequence we show a family of those graphs with an unbounded number of distinct valencies.
In this paper, we completely characterize the graphs with third largest distance eigenvalue at most $-1$ and smallest distance eigenvalue at least $-3$. In particular, we determine all graphs whose distance matrices have exactly two eigenvalues (counting multiplicity) different from $-1$ and $-3$. It turns out that such graphs consist of three infinite classes, and all of them are determined by their distance spectra. We also show that the friendship graph is determined by its distance spectrum.
Signed graphs with exactly two main eigenvalues: The unicyclic case
Shanxi University（山西大学）；Changan University（长安大学）；Hubei Normal University（湖北师范大学）；South China Normal University（华南师范大学）；Shennong Technology Group Co., Ltd.（神农科技集团有限公司）
A mixed graph is obtained by orienting some edges of a simple graph. The positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its Hermitian adjacency matrix, including multiplicities. This matrix was introduced by Liu and Li, independently by Guo and Mohar, in the study of graph energy. Recently, Yuan et al. characterized the mixed graphs with exactly one positive eigenvalue. In this paper, we study the positive inertia indices of mixed graphs and characterize the mixed graphs with cut vertices having positive inertia index 2.
The anti-adjacency matrix of a graph is constructed from the distance matrix of a graph by keeping each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping in each row and each column only the distances equal to 1. The (anti-)adjacency eigenvalues of a graph are those of its (anti-)adjacency matrix. Employing a novel technique introduced by Haemers [Spectral characterization of mixed extensions of small graphs, Discrete Math. 342 (2019) 2760--2764], we…
An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.
An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. It is well known that a graph $G$ has exactly two main eigenvalues if and only if there exists a unique pair of integers $a$ and $b$ such that $\sum_{u\in N(v)}d(u)=ad(v)+b$ for every vertex $v\in V(G)$. We collect such connected graph $G$ in the set $\mathscr{G}(a,b)$. In this paper, we mainly focus to the existence of such $a$ and $b$, and give the necessary and sufficient condition for $\mathscr{G}(a,b)\neq\emptyset$. In addition, we give the bound for the vertex …
In this paper, we characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues of which one is equal to $1$, determine all connected bipartite graphs with at least one vertex of degree $1$ having exactly four distinct normalized Laplacian eigenvalues, and find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.
An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.
It is a well-known fact that a graph of diameter $d$ has at least $d+1$ eigenvalues. Let us call a graph \emph{$d$-extremal} if it has diameter $d$ and exactly $d+1$ eigenvalues. Such graphs have been intensively studied by various authors. %Much attention has been devoted to the study of graphs that are extremal with respect to this relation: \emph{i.e} have diameter $d$ and exactly $d+1$ distinct eigenvalues. A graph is \emph{split} if its vertex set can be partitioned into a clique and a stable set. Such a graph has diameter at most $3$. We obtain a complete classification of the connected …
The eccentricity (anti-adjacency) matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities in each row and each column. This matrix is first defined in 2018 by Wang et al. \cite{1}. In this paper we have characterized the graphs which have at most one (hence exactly) positive eigenvalue of $\varepsilon(G)$.
A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, we first give the necessary and sufficient conditions for a graph with one main signless Laplacian eigenvalue or two main signless Laplacian eigenvalues, and then characterize the trees and unicyclic graphs with exactly two main signless Laplacian eigenvalues, respectively.
A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this paper we give some sufficient and necessary conditions for a Neumaier graph to be strongly regular. Further we show that there does not exist Neumaier graphs with exactly four distinct eigenvalues. We also determine the Neumaier graphs with smallest eigenvalue -2.
A well known upper bound for the spectral radius of a graph, due to Hong, is that $μ_1^2 \le 2m - n + 1$. It is conjectured that for connected graphs $n - 1 \le s^+ \le 2m - n + 1$, where $s^+$ denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete $q$-partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general.
In 1974 Cvetković and Simić showed which graphs $G$ are the bipartite complements of line graphs. In 2002 Borovićanin showed which line graphs $L(H)$ have third largest eigenvalue $λ_3\leq0$. Our first observation is that two of the graphs Borovićanin found are the complements of two of the graphs found by Cvetković and Simić. Using the Courant-Weyl inequalities we show why this is and reprove the result of Borovićanin, highlighting some features of the graphs found by both.
Terpai [22] proved the Nordhaus-Gaddum bound that $μ(G) + μ(\overline{G}) \le 4n/3 - 1$, where $μ(G)$ is the spectral radius of a graph $G$ with $n$ vertices. Let $s^+$ denote the sum of the squares of the positive eigenvalues of $G$. We prove that $\sqrt{s^{+}(G)} + \sqrt{s^+(\overline{G})} < \sqrt{2}n$ and conjecture that $\sqrt{s^{+}(G)} + \sqrt{s^+(\overline{G})} \le 4n/3 - 1.$ We have used AutoGraphiX and Wolfram Mathematica to search for a counter-example. We also consider Nordhaus-Gaddum bounds for $s^+$ and bounds for the Randić index.
In this paper, we show that the eigenvalues of certain classes of Cayley graphs are integers. The (n,k,r)-arrangement graph A(n,k,r) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they differ in exactly r positions. We establish a relation between the eigenvalues of the arrangement graphs and the eigenvalues of certain Cayley graphs. As a result, the conjecture on integrality of eigenvalues of A(n,k,1) follows.
It is known that for a totally positive (TP) matrix, the eigenvalues are positive and distinct and the eigenvector associated with the smallest eigenvalue is totally nonzero and has an alternating sign pattern. Here, a certain weakening of the TP hypothesis is shown to yield a similar conclusion.
The Inverse Symplectic Eigenvalue Problem of a Graph
University of Regina（雷吉纳大学）；American Institute of Mathematics（美国数学研究所）；Iowa State University（爱荷华州立大学）；Purdue University（普渡大学）；University of Wyoming（怀俄明大学）；Kutztown University of Pennsylvania（宾夕法尼亚库兹敦大学）
Let $S_n$ denote the symmetric group on $n$ letters. The $k$-point fixing graph $\mathcal{F}(n,k)$ is defined to be the graph with vertex set $S_n$ and two vertices $g,h$ of $\mathcal{F}(n,k)$ are joined by an edge, if and only if $gh^{-1}$ fixes exactly $k$ points. Ku, Lau and Wong [Cayley graph on symmetric group generated by elements fixing $k$ points, Linear Algebra Appl. 471 (2015) 405-426] obtained a recursive formula for the eigenvalues of $\mathcal{F}(n,k)$. In this paper, we use objects called excited diagrams defined as certain generalizations of skew shapes and derive an explicit fo…
In this paper we study the positive definiteness of meet and join matrices using a novel approach. When the set $S_n$ is meet closed, we give a sufficient and necessary condition for the positive definiteness of the matrix $(S_n)_f$. From this condition we obtain some sufficient conditions for positive definiteness as corollaries. We also use graph theory and show that by making some graph theoretic assumptions on the set $S_n$ we are able to reduce the assumptions on the function $f$ while still preserving the positive definiteness of the matrix $(S_n)_f$. Dual theorems of these results for j…
Answering a question of Jiang and Polyanskii as well as Jiang, Tidor, Yao, Zhang, and Zhao, we show the existence of infinitely many angles $θ$ for which the maximum number of lines in $\mathbb R^n$ meeting at the origin with pairwise angles $θ$ exceeds $n+Ω(\log\log n)$ but is at most $n+o(n)$. To accomplish this, we construct, for various real $λ$ and integer $d$, $d$-regular graphs with second eigenvalue exactly $λ$ and arbitrarily large second eigenvalue multiplicity. Central to our construction is a distribution on factors of bipartite graphs which possesses concentration properties.
Adinkras are signed graphs used to study supersymmetry in physics. We provide an introduction to these objects, and study the properties of their signed adjacency and signed Laplacian matrices. These matrices each have exactly two distinct eigenvalues (of equal multiplicity), making Adinkras closely related to the notions of strongly regular graphs. We also study the critical groups of Adinkras, and in particular determine their odd components. A novel technique of independent interest is used which considers critical groups over polynomial rings.
