# New bound for edge spectral radius and edge energy of graphs

Document Type : Research Paper

Authors

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

10.22075/ijnaa.2021.23361.2523

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 $E_e(X) =\sum_{i=1}^{m} \vert \mu_i \vert$.
Suppose $B^{'}$ be an edge subset of $E(X)$ (set of edges of $X$). For any $e \in B^{'}$ the degree of the edge $e_i$ with respect to the subset $B^{'}$ is defined as the number of edges in $B^{'}$ that are adjacent to $e_i$. We call it as $\varepsilon$-degree and is denoted by $\varepsilon_i$. Denote $\mu_1(X)$ as the largest eigenvalue of the graph $X$ and $s_i$ as the sum of $\varepsilon$-degree of edges that are adjacent to $e_i$. In this paper, we give lower bounds of $\mu_1(X)$ and $\mu_1^{D^{'}}(X)$ in terms of $\varepsilon$-degree. Consequently, some existing bounds on the graph invariants $E_e(X)$ are improved.

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