Related papers: On the largest eigenvalue of a mixed graph with pa…

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…

Given a graph $G$, the adjacency matrix and degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} proposed the $A_{\alpha}$-matrix: $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G),$ where…

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is constructed from the distance matrix of $G$ by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency…

Let $\G$ be an oriented graph of order $n$ and $\a_1,\a_2,..., \a_n$ denote all the eigenvalues of the skew-adjacency matrix of $\G.$ The skew energy $\displaystyle{\cal E}_s(\G)= \sum_{i=1}^{n} |\a_i|.$ In this paper, the oriented…

Given any graph $G$, the (adjacency) spread of $G$ is the maximum absolute difference between any two eigenvalues of the adjacency matrix of $G$. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and…

In this paper, we will give a structure theory for signed graphs with fixed smallest eigenvalue and investigate signed graphs with smallest eigenvalue greater than $-1-\sqrt{2}$. Given a real number $\lambda\leq -1$, we show that the…

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. In this paper, we give a characterization of the star graph,…

This paper examines the spectral characterizations of oriented graphs. Let $\Sigma$ be an $n$-vertex oriented graph with skew-adjacency matrix $S$. Previous research mainly focused on self-converse oriented graphs, proposing arithmetic…

Oriented line graph, introduced by Kotani and Sunada (2000), is closely related to Hashimato's non-backtracking matrix (1989). It is known that for regular graphs $G$, the eigenvalues of the adjacency matrix of the oriented line graph…

For a group $\Gamma$, a $\Gamma$-labelled graph is an undirected graph $G$ where every orientation of an edge is assigned an element of $\Gamma$ so that opposite orientations of the same edge are assigned inverse elements. A path in $G$ is…

A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of…

An instance of the maximum mixed graph orientation problem consists of a mixed graph and a collection of source-target vertex pairs. The objective is to orient the undirected edges of the graph so as to maximize the number of pairs that…

The rank $r(G)$ of a graph $G$ is the rank of its adjacency matrix $A(G)$ and the nullity $\eta(G)$ of $G$ is the multiplicity of $0$ as an eigenvalue of $A(G)$. In this paper, we prove that if $G$ is a connected graph of order $n$ with…

The independence polynomial of a graph $G$, denoted $I(G,x)$, is the generating polynomial for the number of independent sets of each size. The roots of $I(G,x)$ are called the \textit{independence roots} of $G$. It is known that for every…

Let $k$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $\delta$. A graph $G$ is said to have property $P(k, d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with edge…

A stable set in a graph G is a set of mutually non-adjacent vertices, alpha(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T, of order…

Let $G=(V(G), E(G))$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. Let $S$ be a subset of $V(G)$, and let $B(S)$ be the set of neighbours of $S$ in $V(G) \setminus S$. The differential $\partial(S)$ of $S$ is the number…

This paper establishes new upper bounds for the sum of the $k$ largest eigenvalues of symmetric matrices. When applied to the adjacency matrix of a graph, our results improve upon a related bound due to Mohar {\bf [On the sum of k largest…

A {\it semi-proper orientation} of a given graph $G$ is a function $(D,w)$ that assigns an orientation $D(e)$ and a positive integer weight $ w(e)$ to each edge $e$ such that for every two adjacent vertices $v$ and $u$, $S_{(D,w)}(v) \neq…

For a graph $G$, the $\gamma$-graph of $G$, $G(\gamma)$, is the graph whose vertices correspond to the minimum dominating sets of $G$, and where two vertices of $G(\gamma)$ are adjacent if and only if their corresponding dominating sets in…