Related papers: Complete solution to a problem on the maximal ener…

Given a graph $G$, we associate a path matrix $P$ whose $(i, j)$ entry represents the maximum number of vertex disjoint paths between the vertices $i$ and $j$, with zeros on the main diagonal. In this note, we resolve four conjectures from…

Let $ G $ be a simple graph with the vertex cover number $ \tau $. The energy $ \mathcal{E}(G) $ of $ G $ is the sum of the absolute values of all the adjacency eigenvalues of $ G $. In this article, we establish $ \mathcal{E}(G)\geq 2\tau…

The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Some variants of energy can also be found in the literature which are defined on the concepts of Laplacian matrix, Distance…

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

In this note we prove that the vertex energy of a graph, as defined in Arizmendi and Juarez (2018), can be calculated in terms of a Coulson integral formula. We present examples of how this formula can be used, and we show some applications…

For a graph $G$, its energy $\mathcal{E}(G)$ is the sum of absolute values of the eigenvalues of its adjacency matrix, the matching number $\mu(G)$ is the number of edges in a maximum matching of $G$, while $\Delta$ is the maximum vertex…

A signed graph $\Gamma(G)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $\Gamma(G)$. The energy of a signed graph $\Gamma(G)$ is the sum of the absolute values of the eigenvalues of the…

Motivated by the linear time algorithm that locates the eigenvalues of a cograph G [10], we investigate the multiplicity of eigenvalue for \lambda \neq -1,0. For cographs with balanced cotrees we determine explicitly the highest value for…

The energy $E(G)$ of a simple graph $G$ is the sum of absolute values of the eigenvalues of its adjacency matrix. A borderenergetic graph of order $n \in \mathbb{N}$ is any noncomplete graph~$G$ such that $E(G) = E(K_n) = 2n - 2$. Here we…

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,…

In 1978, motivated by E. H\"uckel's work in quantum chemistry, I. Gutman introduced the concept of the energy of a finite simple graph $G$ as the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. At the time of…

The extended adjacency matrix of a graph with $n$ vertices is a real symmetric matrix of order $n\times n$ whose $(i,j)$-th entry is the average of the ratio of the degree of the vertex $i$ to that of the vertex $j$ and its reciprocal when…

For a simple graph $G$ with $n$ vertices, let $A_G$ denote the adjacency matrix of $G$, and let $\lambda_1(G) \geq \lambda_2(G) \geq \dots \geq \lambda_n(G)$ be its eigenvalues. For an integer $p \geq 2$, the positive $p$-energy and…

Let $G$ be a graph on $n$ vertices and $m$ edges. For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of $G$ is defined as $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal…

For a graph $G$, let $S(G)$ be the Seidel matrix of $G$ and $\te_1(G),...,\te_n(G)$ be the eigenvalues of $S(G)$. The Seidel energy of $G$ is defined as $|\te_1(G)|+...+|\te_n(G)|$. Willem Haemers conjectured that the Seidel energy of any…

Let $G_S$ be a graph with loops obtained from a graph $G$ of order $n$ and loops at $S \subseteq V(G)$. In this paper, we establish a neccesary and sufficient condition on the bipartititeness of a connected graph $G$ and the spectrum…

Let $G$ be a simple graph of order $n$ and $\mu_{1},\mu_{2},\ldots,\mu_{n}$ be the roots of its matching polynomial. The matching energy is defined as the sum $\sum^{n}_{i=1}|\mu_{i}|$, which was introduced by Gutman and Wagner in 2012. In…

In his survey "Beyond graph energy: Norms of graphs and matrices" (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph,…

Let $G$ be a simple graph of order $n$ with degree sequence $(d)=(d_1,d_2,\ldots,d_n)$ and conjugate degree sequence $(d^*)=(d_1^*,d_2^*,\ldots,d_n^*)$. In \cite{AkbariGhorbaniKoolenObudi2010,DasMojallalGutman2017} it was proven that…

The energy of a graph is defined as the sum the absolute values of the eigenvalues of its adjacency matrix. A graph G on n vertices is said to be borderenergetic if its energy equals the energy of the complete graph Kn. In this paper, we…