Let $G$ be a graph with order $n$ and adjacency matrix $\mathbf{A}(G)$. The adjacency polynomial of $G$ is defined as $ϕ(G;λ) =det(λ\mathbf{I}-\mathbf{A}(G))=\sum_{i=0}^n\mathbf{a_i}(G)λ^{n-i}$. Hereafter, $\mathbf{a}_i(G)$ is called the $i$-th adjacency coefficient of $G$. Denote by $\mathfrak{G}_{n,m}$ the set of all connected graphs having $n$ vertices and $m$ edges. A graph $G$ is said $4$-Sachs minimal if $$\mathbf{a}_4(G)=min\{\mathbf{a}_4(H)|H\in \mathfrak{G}_{n,m}\}.$$ The value $min\{\mathbf{a}_4(H)|H\in \mathfrak{G}_{n,m}\}$ is called the minimal $4$-Sachs number in $\mathfrak{G}_{n,m}$, denoted by $\bar{\mathbf{a}}_4(\mathfrak{G}_{n,m})$. In this paper, we study the relationship between the value $\mathbf{a}_4(G)$ and its structural properties. Especially, we give a structural characterization on $4$-Sachs minimal graphs, showing that each $4$-Sachs minimal graph contains a difference graph as its spanning subgraph (see Theorem 8). Then, for $n\ge 4$ and $n-1\le m\le 2n-4$, we determine all $4$-Sachs minimal graphs together with the corresponding minimal $4$-Sachs number $\bar{\mathbf{a}}_4(\mathfrak{G}_{n,m})$.
Let $G$ be a simple graph with order $n$ and adjacency matrix $\mathbf{A}(G)$. Let $ϕ(G; λ)=\det(λI-\mathbf{A}(G))=\sum_{i=0}^n\mathbf{a}_i(G)λ^{n-i}$ be the characteristic polynomial of $G$, where $\mathbf{a}_i(G)$ is called the $i$-th adjacency coefficient of $G$. Denote by $\mathfrak{B}_{n,m}$ the set of all connected graphs having $n$ vertices and $m$ edges. A bipartite graph $G$ is referred as bipartite optimal if $$\mathbf{a}_4(G)=min\{\mathbf{a}_4(H)|H\in \mathfrak{B}_{n,m}\}.$$ The value $min\{\mathbf{a}_4(H)|H\in \mathfrak{B}_{n,m}\}$ is called the minimal $4$-Sachs number in $\mathfr…
We study the equivalence between bipartiteness and symmetry of spectra of mixed graphs, for $θ$-Hermitian adjacency matrices defined by an angle $θ\in (0, π]$. We show that this equivalence holds when, for example, an angle $θ$ is an algebraic number, while it breaks down for any angle $θ\in \mathbb{Q}π$. Furthermore, we construct a family of non-bipartite mixed graphs having the symmetric spectra for given $θ\in \mathbb{Q}π$.
There are typically several nonisomorphic graphs having a given degree sequence, and for any two degree sequence terms it is often possible to find a realization in which the corresponding vertices are adjacent and one in which they are not. We provide necessary and sufficient conditions for two vertices to be adjacent (or nonadjacent) in every realization of the degree sequence. These conditions generalize degree sequence and structural characterizations of the threshold graphs, in which every adjacency relationship is forcibly determined by the degree sequence. We further show that degree se…
The weighted adjacency matrix $A_{f}(G)$ of a simple graph $G=(V,E)$ is the $|V|\times|V|$ matrix whose $ij$-entry equals $f(d_{i},d_j)$, where $f(x,y)$ is a symmetric function such that $f(d_i,d_j)>0$ if $ij\in E$ and $f(d_i,d_j)=0$ if $ij\notin E$ and $d_i$ is the degree of the vertex $i$. In this paper, we determine the unique graph having the largest spectral radius of $A_{f}(G)$ among all the bicyclic graphs under the assumption that $f(x,y)$ is increasing and convex in $x$ and $f(x_1,y_1)\geq f(x_2,y_2)$ when $|x_1-y_1|>|x_2-y_2|$ and $x_1+y_1=x_2+y_2$. Moreover, we determine the unique …
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 $\varepsilon(G)$.
HyperBagGraphs (hb-graphs as short) extend hypergraphs by allowing the hyperedges to be multisets. Multisets are composed of elements that have a multiplicity. When this multiplicity has positive integer values, it corresponds to non ordered lists of potentially duplicated elements. We define hb-graphs as family of multisets over a vertex set; natural hb-graphs correspond to hb-graphs that have multiplicity function with positive integer values. Extending the definition of e-adjacency to natural hb-graphs, we define different way of building an e-adjacency tensor, that we compare before having…
This paper introduces a new graph construction, the permutational power of a graph, whose adjacency matrix is obtained by the composition of a permutation matrix with the adjacency matrix of the graph. It is shown that this construction recovers the classical zig-zag product of graphs when the permutation is an involution, and it is in fact more general. We start by discussing necessary and sufficient conditions on the permutation and on the adjacency matrix of a graph to guarantee their composition to represent an adjacency matrix of a graph, then we focus our attention on the cases in which …
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.
It is well known that spectral Turán type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Turán type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we say $G$ is \textit{$\mathcal{G}$-free} if $G$ does not contain any element of $\mathcal{G}$ as a subgraph. Denote by $λ_1$ and $λ_2$ the largest and the second largest eigenvalues of the adjacency matrix $A(G)$ of $G,$ respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. …
The adjacency matrices of graphs form a special subset of the set of all integer symmetric matrices. The description of which graphs have all their eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most 2) has been known for several decades. In 2007 we extended this classification to arbitrary integer symmetric matrices. In this paper we turn our attention to symmetrizable matrices. We classify the connected nonsymmetric but symmetrizable matrices which have entries in $\Z$ that are maximal with respect to having all their eigenvalues in [-2,2]. This includes a spectral…
In this paper we construct distance-regular graphs admitting a transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$ and $M_{24}$. From the code spanned by the adjacency matrix of the strongly regular graph with parameters (176,70,18,34) we obtain block designs having the full automorphism groups isomorphic to the Higman-Sims finite simple group. Further, we discuss a possibility of permutation decoding of the codes spanned by the adjacency matrices of the graphs constructed and find small PD-sets for some of th…
Let $G$ be a graph with adjacency matrix $A$. The transition matrix corresponding to $G$ is defined by $H(t):=\exp{\left(itA\right)}$, $t\in\Rl$. The graph $G$ is said to have perfect state transfer (PST) from a vertex $u$ to another vertex $v$, if there exist $τ\in\Rl$ such that the $uv$-th entry of $H(τ)$ has unit modulus. The graph $G$ is said to be periodic at $τ\in\Rl$ if there exist $γ\in\Cl$ with $|γ|=1$ such that $H(τ)=γI$, where $I$ is the identity matrix. A $\mathit{gcd}$-graph is a Cayley graph over a finite abelian group defined by greatest common divisors. In this paper, we constr…
The index of a graph is the largest eigenvalue of its adjacency matrix. A starlike is a tree having a unique vertex of degree $r>2$. We show how to order the starlike trees with $n>3$ by their indices. In particular, the indices of starlike trees are all distinct.
Let $G$ be a simple undirected graph, and $G^ϕ$ be a mixed graph of $G$ with the generalized orientation $ϕ$ and Hermitian-adjacency matrix $H(G^ϕ)$. Then $G$ is called the underlying graph of $G^ϕ$. The Hermitian energy of the mixed graph $G^ϕ$, denoted by $\mathcal{E}_H(G^ϕ)$, is defined as the sum of all the singular values of $H(G^ϕ)$. A $k$-regular mixed graph on $n$ vertices having Hermitian energy $n\sqrt{k}$ is called a $k$-regular optimum Hermitian energy mixed graph. In this paper, we first focus on the problem proposed by Liu and Li [J. Liu, X. Li, Hermitian-adjacency matrices and H…
Small But Unwieldy: A Lower Bound on Adjacency Labels for Small Classes
Univ. Lyon, ENS de Lyon, UCBL, CNRS, LIP（里昂大学，里昂高等师范学院，里昂第一大学，法国国家科学研究中心，信息处理实验室）；Department of Computer Science, University of Liverpool（利物浦大学计算机系）；Department of Computer Science, University of Sheffield（谢菲尔德大学计算机系）
Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov [Appl. Anal. Discrete Math., 11 (2017) 81--107] defined the matrix $A_α(G) = αD(G) + (1-α)A(G)$ for any real $α\in[0,1]$. The largest eigenvalue of $A(G)$ is called the spectral radius of $G$, while the largest eigenvalue of $A_α(G)$ is called the $A_α$ spectral radius of $G$. Let $\mathcal{G}_{n,i}$ be the set of graphs of order $n$ with independence number $i$. Recently, for all graphs in $\mathcal{G}_{n,i}$ having the minimum or the maximum $A$, $Q$ and $A_α$ spectral radius where $i\in\{1…
The main eigenvalues of a graph $G$ are those eigenvalues of the $(0,1)$-adjacency matrix $\mathbf A$ having a corresponding eigenvector not orthogonal to $\mathbf j = (1,\dots,1)$. The CDC of a graph $G$ is the direct product $G\times K_2$. The main eigenspace of $\mathbf A$ is generated by the principal main eigenvectors and is the same as the image of the walk matrix. A hierarchy of properties of pairs of graphs is established in view of their CDC's, walk matrices, main eigenvalues, eigenvectors and eigenspaces. We determine by algorithm that there are 32 pairs of non-isomorphic graphs on a…
