Document Type : Research Paper

Authors

• Saeed Mohammadian Semnani
• Samira Sabeti

Department of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran.

Abstract

Let $X(V,E)$ be a simple graph with $n$ vertices and $m$ edges without isolated vertices. Denote by $B=(b_{ij}{)}_{m\times m}$ the edge adjacency matrix of $X$. Eigenvalues of the matrix $B$, ${\mu }_1,{\mu }_2,\cdots ,{\mu }_m$, are the edge spectrum of the graph $X$. An important edge spectrum-based invariant is the graph energy, defined as $E_e(X)=\sum _{i=1}^m|{\mu }_i|$. Suppose $B^{{}^{\prime }}$ be an edge subset of $E(X)$ (set of edges of $X$). For any $e\in B^{{}^{\prime }}$ the degree of the edge $e_i$ with respect to the subset $B^{{}^{\prime }}$ is defined as the number of edges in $B^{{}^{\prime }}$ that are adjacent to $e_i$. We call it as $\epsilon$-degree and is denoted by ${\epsilon }_i$. Denote ${\mu }_1(X)$ as the largest eigenvalue of the graph $X$ and $s_i$ as the sum of $\epsilon$-degree of edges that are adjacent to $e_i$. In this paper, we give lower bounds of ${\mu }_1(X)$ and ${\mu }_1^{D^{{}^{\prime }}}(X)$ in terms of $\epsilon$-degree. Consequently, some existing bounds on the graph invariants $E_e(X)$ are improved.

Keywords

• ε-degree
• adjacency matrix
• spectral radius
• dominating set
• graph energy
• bound of energy