Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 13 1 2022 03 01 New bound for edge spectral radius and edge energy of graphs 1175 1181 5661 10.22075/ijnaa.2021.23361.2523 EN Saeed Mohammadian Semnani Department of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran. Samira Sabeti Department of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran. Journal Article 2021 05 08 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. https://ijnaa.semnan.ac.ir/article_5661_ce621f9a383e05b56c5982ef6747d6af.pdf