Related papers: On the eigenvalues of some signed graphs

If alpha=alpha(G) is the maximum size of an independent set and s_{k} equals the number of stable sets of cardinality k in graph G, then I(G;x)=s_{0}+s_{1}x+...+s_{alpha}x^{alpha} is the independence polynomial of G. In this paper we prove…

A symmetric matrix $M=(m_{ij}) \in \mathbb{R}^{n \times n}$ is said to be associated with an $n$-vertex graph $G=(V,E)$ with vertex set $\{v_1,\ldots,v_n\}$ if, for every $i \neq j$, we have $m_{ij} \neq 0$ if and only if $\{v_i,v_j\}\in…

Let $G$ be a graph of order $n$ and $\mu$ be an adjacency eigenvalue of $G$ with multiplicity $k\geq 1$. A star complement $H$ for $\mu$ in $G$ is an induced subgraph of $G$ of order $n-k$ with no eigenvalue $\mu$, and the vertex subset…

In this brief communication, we investigate the cospectral as well integral chain graphs for Seidel matrix, a key component to study the structural properties of equiangular lines in space. We derive a formula that allows to generate an…

Two signed graphs are called switching isomorphic if one of them is isomorphic to a switching equivalent of the other. To determine the number of switching non-isomorphic signed graphs on a specific graph, we will establish a method based…

Let $G$ be a connected graph on $n$ vertices and $d_{ij}$ be the length of the shortest path between vertices $i$ and $j$ in $G$. We set $d_{ii}=0$ for every vertex $i$ in $G$. The squared distance matrix $\Delta(G)$ of $G$ is the $n\times…

For any graph $G$ on $n$ vertices and for any {\em symmetric} subgraph $J$ of $K_{n,n}$, we construct an infinite sequence of graphs based on the pair $(G,J)$. The First graph in the sequence is $G$, then at each stage replacing every…

Let $G$ be a simple, connected graph and let $A(G)$ be the adjacency matrix of $G$. If $D(G)$ is the diagonal matrix of the vertex degrees of $G$, then for every real $\alpha \in [0,1]$, the matrix $A_{\alpha}(G)$ is defined as…

For a positive integer $k\ge 1$, a graph $G$ is $k$-stepwise irregular ($k$-SI graph) if the degrees of every pair of adjacent vertices differ by exactly $k$. Such graphs are necessarily bipartite. Using graph products it is demonstrated…

A square (0,1)-matrix X of order n > 0 is called fully indecomposable if there exists no integer k with 0 < k < n, such that X has a k by n-k zero submatrix. A stable set of a graph G is a subset of pairwise nonadjacent vertices. The…

The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts.…

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. We consider the eigenvalues of adjacency matrices of cographs and prove that a graph $G$ is a cograph if and only if no induced subgraph of $G$ has an…

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…

The idiosyncratic polynomial of a graph $G$ with adjacency matrix $A$ is the characteristic polynomial of the matrix $ A + y(J-A-I)$, where $I$ is the identity matrix and $J$ is the all-ones matrix. It follows from a theorem of Hagos (2000)…

An independent set in a graph is a set of pairwise non-adjacent vertices, and alpha(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while mu(G) is the cardinality of a maximum matching.…

We study eigenvalue distribution of the adjacency matrix $A^{(N,p, \alpha)}$ of weighted random bipartite graphs $\Gamma= \Gamma_{N,p}$. We assume that the graphs have $N$ vertices, the ratio of parts is $\frac{\alpha}{1-\alpha}$ and the…

It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…

The $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G=(V,E)$ is a quantitative measure of its $d$-dimensional rigidity, defined in terms of the eigenvalues of stiffness matrices associated with different embeddings of the graph…

We investigate the distribution of eigenvalues of weighted adjacency matrices from a specific ensemble of random graphs. We distribute $N$ vertices across a fixed number $\kappa$ of components, with asymptotically $\alpha_j \dot N$ vertices…

Let $G$ be an $n$-vertex graph with adjacency matrix $A$, and $W=[e,Ae,\ldots,A^{n-1}e]$ be the walk matrix of $G$, where $e$ is the all-one vector. In Wang [J. Combin. Theory, Ser. B, 122 (2017): 438-451], the author showed that any graph…