Related papers: Hermitian adjacency matrix of the second kind for …

A mixed graph $\widetilde{G}$ is obtained by orienting some edges of a graph $G$, where $G$ is the underlying graph of $\widetilde{G}$. Let $r(\widetilde{G})$ be the $H$-rank of $\widetilde{G}$. Denote by $r(G)$, $\kappa(G)$, $m(G)$ and…

We introduce a transfer matrix method for the spectral analysis of discrete Hermitian operators with locally finite hopping. Such operators can be associated with a locally finite graph structure and the method works in principle on any…

A finite graph $\Gamma$ is called $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. We study a family of symmetric graphs, called the unitary…

Let $H$ be a connected $m$-uniform hypergraph, and let $\mathcal{A}(H)$ be the adjacency tensor of $H$ whose spectrum is simply called the spectrum of $H$. Let $s(H)$ denote the number of eigenvectors of $\mathcal{A}(H)$ associated with the…

Graph edit distance / similarity is widely used in many tasks, such as graph similarity search, binary function analysis, and graph clustering. However, computing the exact graph edit distance (GED) or maximum common subgraph (MCS) between…

This work considers the distributed computation of the one-to-one vertex correspondences between two undirected and connected graphs, which is called \textit{graph matching}, over multi-agent networks. Given two \textit{isomorphic} and…

The joint degree matrix of a graph gives the number of edges between vertices of degree i and degree j for every pair (i,j). One can perform restricted swap operations to transform a graph into another with the same joint degree matrix. We…

Let $\gamma$ be a constant such that $0 < \gamma < 1$, and let $n$ be a sufficiently large integer. Consider a $3$-uniform hypergraph $H$ on $n$ vertices. In 2013, K\"{u}hn, Osthus, and Treglown, along with Khan independently, proved that…

The universal adjacency matrix $U$ of a graph $\Gamma$, with adjacency matrix $A$, is a linear combination of $A$, the diagonal matrix $D$ of vertex degrees, the identity matrix $I$, and the all-1 matrix $J$ with real coefficients, that is,…

For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{i}$ the degree of vertex $v_{i}$. Let $f(x, y)>0$ be a real symmetric function in $x$ and $y$. The weighted adjacency matrix $A_{f}(G)$ of a graph $G$ is a square matrix, where the…

A mixed graph is said to be HS-\emph{integral} if the eigenvalues of its Hermitian-adjacency matrix of the second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are…

In this article we consider the graph alignment problem from the perspective of high-dimensional statistics: we aim to estimate an unknown permutation $\pi^*$ from the observation of two correlated random adjacency matrices $A_1$, $A_2$. We…

For a graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A\in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ be the minimum number…

It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to…

We study the balance of $G$-gain graphs, where $G$ is an arbitrary group, by investigating their adjacency matrices and their spectra. As a first step, we characterize switching equivalence and balance of gain graphs in terms of their…

The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of identifying symmetric signings of matrices with natural spectral…

Unitary graphs are arc-transitive graphs with vertices the flags of Hermitian unitals and edges defined by certain elements of the underlying finite fields. They played a significant role in a recent classification of a class of…

The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. It is a known result…

For a graph $G$ of order $n$, let $$ \lambda_1(G)\ge \cdots \ge \lambda_n(G) $$ be the eigenvalues of its adjacency matrix. We prove that every graph $G$ on $n\ge 3$ vertices satisfies $$ \lambda_3(G)\le \frac{n}{3}-1, $$ thereby solving a…

The difference between the two largest eigenvalues of the adjacency matrix of a graph $G$ is called the spectral gap of $G.$ If $G$ is a regular graph, then its spectral gap is equal to algebraic connectivity. Abdi, Ghorbani and Imrich, in…