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 = (bij)m×m the edge adjacency matrix of x. eigenvalues of the matrix b, µ1, µ2, · · · , µm, are the edge spectrum of the graph x. an important edge spectrum-based invariant is the graph energy, defined as ee(x) = σm i=1|µi|. suppose bʹ be an edge subset of e(x) (set of edges of x). for any e ∈ b 0 the degree of the edge ei with respect to the subset b 0 is defined as the number of edges in bʹ that are adjacent to ei . we call it as ε-degree and is denoted by εi . denote µ1(x) as the largest eigenvalue of the graph x and si as the sum of ε-degree of edges that are adjacent to ei . in this paper, we give lower bounds of µ1(x) and µ dʹ1 (x) in terms of ε-degree. consequently, some existing bounds on the graph invariants ee(x) are improved.