Related papers: Bounds on Path Energy of Graphs

Let $G$ be a graph with the vertex set $ \lbrace v_1,\ldots,v_n \rbrace$. The Seidel matrix of $G$ is an $n\times n$ matrix whose diagonal entries are zero, $ij$-th entry is $-1$ if $ v_{i} $ and $ v_{j} $ are adjacent and otherwise is $ 1…

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and leaving zeros in the remaining ones. The eccentricity energy of $G$ is…

Let $G $ be a graph on $p$ vertices with adjacency matrix $A(G)$ and degree matrix $D(G)$. For each $\alpha \in [0, 1]$, the $A_\alpha$-matrix is defined as $A_\alpha (G) = \alpha D(G) + (1 - \alpha)A(G)$. In this paper, we compute the…

Let $G=(V,E)$ be a simple connected graph. A matching of $G$ is a set of disjoint edges of $G$. For every $n, m\in\mathbb{N}$, the $n$-subdivision of $G$ is a simple graph $G^{\frac{1}{n}}$ which is constructed by replacing each edge of $G$…

In this correspondence, we introduced the concept of minimum roman dominating distance energy $E_{RDd}(G)$ of a graph $G$ and computed minimum roman dominating distance energy of some standard graphs. Also, we discussed the properties of…

This paper explores the Harmonic matrix $MH(G)$ associated with a simple graph $ G $, where each entry corresponds to $ \frac{2}{d_i + d_j} $ for adjacent vertices $ v_i $ and $ v_j $. We investigate the spectral properties of this matrix,…

The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph $G$, is denoted by $q(G)$. Using other parameters related to $G$, bounds for $q(G)$ are proven and then applied to deduce…

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of the eigenvalues of $G$. An $n$-vertex graph is said to be hypoenergetic if $E(G)<n$ and strongly hypoenergetic if $E(G)<n-1$. In this paper, we consider…

We study the graph energy from a cooperative game viewpoint. We introduce \emph{the graph energy game} and show various properties. In particular, we see that it is a superadditive game and that the energy of a vertex, as defined in…

For a graph G, M(G) denotes the maximum multiplicity occurring of an eigenvalue of a symmetric matrix whose zero-nonzero pattern is given by edges of G. We introduce two combinatorial graph parameters T^-(G) and T^+(G) that give a lower and…

This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…

Let $x_1, x_2, \dots, x_n$ be the eigenvalues of a signed graph $\Gamma$ of order $n$. The energy of $\Gamma$ is defined as $E(\Gamma)=\sum^{n}_{j=1}|x_j|.$ Let $\mathcal{P}_n^4$ be obtained by connecting a vertex of the negative circle…

For a simple, undirected and connected graph $G$, $D_{\alpha}(G) = \alpha Tr(G) + (1-\alpha) D(G)$ is called the $\alpha$-distance matrix of $G$, where $\alpha\in [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex…

Gutman and Wagner proposed the concept of the matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let $G$ be a simple graph of order $n$ and $\mu_1,\mu_2,\ldots,\mu_n$ be the roots of its matching…

Many natural and social systems develop complex networks, that are usually modelled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semi-circle law is known to…

We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order. This bound is best possible up to a constant factor and improves prevoius results of…

For a simple graph $G$ with adjacency matrix $A(G)$, let $\pi(G,x):=\mathrm{per}(xI-A(G))$ be its permanental polynomial with roots $\mu_1,\ldots,\mu_n \in \mathbb{C}$, and define the permanental energy $E_{\mathrm{per}}(G):=\sum_{i=1}^n…

We prove that, for any graph $G$, its graph energy is at least twice the Randic index. We show that equality holds if and only if $G$ is the union of complete bipartite graphs.

Let $G$ be a graph with $n$ non-isolated vertices and $m$ edges. The positive / negative square energies of $G$, denoted $s^+(G)$ / $s^-(G)$, are defined as the sum of squares of the positive / negative eigenvalues of the adjacency matrix…

We study the energy per vertex in regular graphs. For every k, we give an upper bound for the energy per vertex of a k-regular graph, and show that a graph attains the upper bound if and only if it is the disjoint union of incidence graphs…