Related papers: Graph Energies of Generalized and Shadow-Splitting…

We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We…

In a graph, we assign distinct integers to the vertices, and take the sum of two integers if they are on two adjacent vertices. The minimum possible number of different sums is the \emph{sum index} of this graph. In this paper, we present…

The power graph $P(\Omega)$ of a group $\Omega$ is a graph with the vertex set $\Omega$ such that two distinct vertices form an edge if and only if one of them is an integral power of the other. In this article, we determine the power graph…

For a simple graph $G$, the generalized adjacency matrix $A_{\alpha}(G)$ is defined as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G), \alpha\in [0,1]$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees of…

Let $n,k$ be positive integers such that $n\geq 3$, $k < \frac {n}{2} $. Let $q$ be a power of a prime $p$ and $\mathbb{F}_q$ be a finite field of order $q$. Let $V(q,n)$ be a vector space of dimension $n$ over $\mathbb{F}_q$. We define the…

For a connected graph $G$, we present the concept of a new graph matrix related to its distance and Seidel matrix, called distance Seidel matrix $\mathcal{D}^S(G)$. Suppose that the eigenvalues of $\mathcal{D}^S(G)$ be $\partial_{1}^{S}(G)…

Let $G$ be a finite group. A number of graphs with the vertex set $G$ have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful…

A $\mathbb{T}$-gain graph, $\Phi = (G, \varphi)$, is a graph in which the function $\varphi$ assigns a unit complex number to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency…

Harshitha et al. recently introduced Seidel energy of graphs with self loops. In this paper, we extend some of their results by giving a necessary and sufficient condition for the Seidel energy of a looped graph to be equal to the Seidel…

Any directed graph G with N vertices and J edges has an associated line-graph L(G) where the J edges form the vertices of L(G). We show that the non-zero eigenvalues of the adjacency matrices are the same for all graphs of such a family…

In this note we elaborate on some notions of surface area for discrete graphs which are closely related to the inverse degree. These notions then naturally lead to associated connectivity measures of graphs and to the definition of a…

Let $p, k, q$ be positive integers with $p-2 \geqslant k$ and let $K_{p,k}^{q}$ be the generalized pineapple graph which is obtained by joining independent set of $q$ vertices with $k$ vertices of a complete graph $K_{p}.$ In \cite{TSH2},…

The energy of a vertex $v_i$ in a graph $G$ is defined as $\mathcal{E}_G(v_i) = |A|_{ii}$, where $A$ is the adjacency matrix of $G$, $A^*$ denotes the conjugate transpose of $A$, and $|A| = (AA^*)^{1/2}$. The total energy of the graph,…

Let $S(G^\sigma)$ be the skew-adjacency matrix of an oriented graph $G^\sigma$. The skew energy of $G^\sigma$ is defined as the sum of all singular values of its skew-adjacency matrix $S(G^\sigma)$. In this paper, we first deduce an…

Let $k\ge 1$ be an odd integer, $t=\lfloor {{k+2}\over 4}\rfloor$, and $q$ be a prime power. We construct a bipartite, $q$-regular, edge-transitive graph $C\!D(k,q)$ of order $v \le 2q^{k-t+1}$ and girth $g \ge k+5$. If $e$ is the the…

The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and…

Here, we define a subdivision operation for a hypergraph and compute all the eigenvalues of the subdivision of regular and certain non-regular hypergraphs. In non-regular hypergraphs, we investigate the power of regular graphs, various…

In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended $p$-sum, or NEPS) of signed graphs. We express the adjacency matrix of the product in terms of…

We introduce the family of $k$-gap-planar graphs for $k \geq 0$, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most $k$ of its crossings. This definition is…

We investigate connections between the symmetries (automorphisms) of a graph and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial automorphism $\phi$, it is possible to use $\phi$ to decompose any matrix…