On graph energy, maximum degree and vertex cover number

For a simple graph $G$ with $n$ vertices and $m$ edges having adjacency eigenvalues $\lambda_1,\lambda_2, \dots,\lambda_n$, the energy $E(G)$ of $G$ is defined as $E(G)=\sum_{i=1}^{n} |\lambda_i|$. We obtain the upper bounds for $E(G)$ in terms of the vertex covering number $\tau$, the number of edges $m$, maximum vertex degree $d_1$ and second maximum vertex degree $d_2$ of the connected graph $G$. These upper bounds improve some recently known upper bounds for $E(G)$. Further, these upper bounds for $E(G)$ imply a natural extension to other energies like distance energy and Randi\'{c} energy associated to a connected graph $G$

A lower bound for the energy of symmetric matrices and graphs

The energy of a symmetric matrix is the sum of the absolute values of its eigenvalues. We introduce a lower bound for the energy of a symmetric partitioned matrix into blocks. This bound is related to the spectrum of its quotient matrix. Furthermore, we study necessary conditions for the equality. Applications to the energy of the generalized composition of a family of arbitrary graphs are obtained. A lower bound for the energy of a graph with a bridge is given. Some computational experiments are presented in order to show that, in some cases, the obtained lower bound is incomparable with the well known lower bound $2\sqrt{m}$, where $m$ is the number of edges of the graph

Energy and laplacian energy of graphs related to a family of finite groups

the energy of a graph is the sum of the absolute value of the eigenvalues of the adjacency matrix of the graph. this quantity is studied in the context of spectral graph theory. the energy of graph was first defined by gutman in 1978. however, the motivation for the study of the energy comes from chemistry, dating back to the work by hukel in the 1930s, where it is used to approximate the total n-electron energy of molecules. recently, the energy of the graph has become an area of interest to many mathematicians and several variations have been introduced. in this research, new theoretical results on the energy and the laplacian energy of some graphs associated to three types of finite groups, which are dihedral groups, generalized quaternion groups and quasidihedral groups are presented. the main aim of this research is to find the energy and laplacian energy of these graphs by using the eigenvalues and the laplacian eigenvalues of the graphs respectively. the results in this research revealed more properties and classifications of dihedral groups, generalized quaternion groups and quasidihedral groups in terms of conjugacy classes of the elements of the groups. the general formulas for the energy and laplacian energy of the conjugacy class graph of dihedral groups, generalized quaternion groups and quasidihedral groups are introduced by using the properties of conjugacy classes of finite groups and the concepts of a complete graph. moreover, the general formula for the energy of the non-commuting graph of these three types of groups are introduced by using some group theory concepts and the properties of the complete multipartite graph. furthermore, the formulas for the laplacian spectrum of the non-commuting graph of dihedral groups, generalized quaternion groups and quasidihedral groups are also introduced, where the proof of the formulas comes from the concepts of an ac-group and the complement of the graph. graphs associated to the relative commutativity degree of subgroups of some dihedral groups are found as the complete multipartite graphs. some formulas for the characteristic polynomials of the adjacency matrices of the graph associated to the relative commutativity degree of subgroups of some dihedral groups and its generalization for some cases depending on the divisors are presented. finally, the energy and the laplacian energy of these graphs for some dihedral groups are computed

New Bounds for the Harmonic Energy and Harmonic Estrada index of Graphs

Let $G$ be a finite simple undirected graph with $n$ vertices and $m$ edges. The Harmonic energy of a graph $G$, denoted by $\mathcal{H}E(G)$, is defined as the sum of the absolute values of all Harmonic eigenvalues of $G$. The Harmonic Estrada index of a graph $G$, denoted by $\mathcal{H}EE(G)$, is defined as $\mathcal{H}EE=\mathcal{H}EE(G)=\sum_{i=1}^{n}e^{\gamma_i},$ where $\gamma_1\geqslant \gamma_2\geqslant \dots\geqslant \gamma_n$ are the $\mathcal{H}$-$eigenvalues$ of $G$. In this paper we present some new bounds for $\mathcal{H}E(G)$ and $\mathcal{H}EE(G)$ in terms of number of vertices, number of edges and the sum-connectivity index