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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