New bound for edge spectral radius and edge energy of graphs

Let $ X(V,E) $ be a simple graph with $ n $ vertices and $ m $ edges without isolated vertices. Denote by $ B = (b_{ij})_{mtimes 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.