On the Randić incidence energy of graphs

Let G= (V, E) , V= { v1, v2, … , vn} , be a simple connected graph with n vertices, m edges and vertex degree sequence Δ= d1≥ d2≥ ⋯ ≥ dn= δ> 0 , di= d(vi). Denote by A=(aij)n×n and D= diag (d1, d2, … , dn) , the adjacency and the diagonal degree matrix of G, respectively. The signless Laplacian of G is defined as L+= D+ A, and the normalized signless Laplacian matrix as L+= D- 1 / 2L+D- 1 / 2= I+ D- 1 / 2AD- 1 / 2. The Randić incidence energy of G is defined as IRE(G)=∑i=1nγi+ , where 2=γ1+≥γ2+≥⋯≥γn+≥0, are eigenvalues of L+. Upper and lower bounds on IRE(G) are obtained.