Related papers: On the eigenvalues of some signed graphs

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…

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…

In this article we investigate normalized adjacency eigenvalues (simply normalized eigenvalues) and normalized adjacency energy of connected threshold graphs. A threshold graph can always be represented as a unique binary string. Certain…

A mixed extension of a graph $G$ is a graph $H$ obtained from $G$ by replacing each vertex of $G$ by a clique or a coclique, whilst two vertices in $H$ corresponding to distinct vertices $x$ and $y$ of $G$ are adjacent whenever $x$ and $y$…

The signless Laplacian matrix of a graph $G$ is given by $Q(G)=D(G)+A(G)$, where $D(G)$ is a diagonal matrix of vertex degrees and $A(G)$ is the adjacency matrix. The largest eigenvalue of $Q(G)$ is called the signless Laplacian spectral…

The characteristic polynomial and the spectrum of the lexicographic product of graphs $H[G]$, a specific instance of the generalized composition (also called $H$-join), are explicitly determined for arbitrary graphs $H$ and $G$, in terms of…

The inertia of a graph $G$ is defined to be the triplet $In(G) = (p(G), n(G), $ $\eta(G))$, where $p(G)$, $n(G)$ and $\eta(G)$ are the numbers of positive, negative and zero eigenvalues (including multiplicities) of the adjacency matrix…

We prove a general upper bound on the $k$-th adjacency eigenvalue of a graph. For $k\ge 2$, we show that \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1 \] for every graph $G$ on $n$ vertices. We build on a recent approach that…

We investigate when a complete graph $K_n$ with some edges deleted is determined by its adjacency spectrum. It is shown to be the case if the deleted edges form a matching, a complete graph $K_m$ provided $m \leq n-2$, or a complete…

Graph is an abstract representation commonly used to model networked systems and structure. In problems across various fields, including computer vision and pattern recognition, and neuroscience, graphs are often brought into comparison (a…

Let $G$ be a bipartite graph and its adjacency matrix $\mathbb A$. If $G$ has a unique perfect matching, then $\mathbb A$ has an inverse $\mathbb A^{-1}$ which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph.…

Let $G$ be a connected simple graph with $n$ vertices. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$.…

In this paper we study two classes of graphs, the (m,k)-stars and l-dependent graphs, investigating the relation between spectrum characteristics and graph structure: conditions on the topology and edge weights are given in order to get…

The parameter $q(G)$ of a graph $G$ is the minimum number of distinct eigenvalues of a symmetric matrix whose pattern is given by $G$. We introduce a novel graph product by which we construct new infinite families of graphs that achieve…

Let \Gamma be a signed graph and let A(\Gamma) be the adjacency matrix of \Gamma. The nullity of \Gamma is the multiplicity of eigenvalue zero in the spectrum of A(\Gamma). In this paper we characterize the signed graphs of order n with…

Let mu(G) and mu_min(G) be the largest and smallest eigenvalues of the adjacency matricx of a graph G. We refine quantitatively the following two results on graph spectra. (i) if H is a proper subgraph of a connected graph G, then…

In this paper we give a new generalization of token graphs. Given two integers $1\leq m \leq k$ and a graph $G$ we define the generalized token graph of the graph $G$, to be the graph $F_k^m(G)$ whose vertices correspond to configurations…

We study the algebraic connectivity (or second Laplacian eigenvalue) of token graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$,…

Given a graph $G$, the graph $[G]$ obtained by adding, for each pair of vertices of $G$, a unique vertex adjacent to both vertices is called the binding graph of $G$. In this work, we show that the class of binding graphs is…

For a set W of vertices and a vertex v in a graph G, the k-vector r2(v|W) = (aG(v,w1),...,aG(v,wk)) is the adjacency representation of v with respect to W, where W = {w1,...,wk} and aG(x,y) is the minimum of 2 and the distance between the…