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)=∑mi=1|μi|. Suppose B′ be an edge subset of E(X) (set of edges of X). For any e∈B′ the degree of the edge ei with respect to the subset B′ is defined as the number of edges in B′ 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 μD′1(X) in terms of ε-degree. Consequently, some existing bounds on the graph invariants Ee(X) are improved.