Some new inequalities on the normalized signless Laplacian resolvent energy of graphs

For a connected graph G of order n with normalized signless Laplacian eigenvalues γ1+ ≥ γ2+ ≥ ··· ≥ γn+ ≥ 0, the normalized signless Laplacian resolvent energy of G is defined as ERNS(G) = Pni=1 3−γi + . In this paper, we derive new inequalities on 1 ERNS(G) as well as relations on ERNS(G) with Randić (normalized) incidence energy. We also deduced that some of our results improve the existing results in the literature.