Comparison of Resolvent Energies of Laplacian Matrices

Let λ₁ ≥ λ₂ ≥ · · · ≥ λ_n be the eigenvalues of the adjacency matrix of a simple graph G of order n. A graph-spectrum-based invariant, resolvent energy, put forward by Gutman et al. [Resolvent energy of graphs, MATCH Commun. Math. Comput. Chem. 75 (2016) 279–290], is defined as ER(G) =^∑n i=1^(n−λi)⁻¹. After that two more resolvent energies defined in the literature, first one is Laplacian resolvent energy (RL) and the second one is signless Laplacian resolvent energy (RQ). In this paper we define normalized Laplacian resolvent energy (ERN), and give some lower and upper bounds on ER, RL and ERN of graphs, and characterize the extremal graphs. In particular, we obtain some relations between Laplacian resolvent energy (RL) with popular graph invariants, like Kirchhoff index and the number of spanning trees of graphs. Moreover we compare between resolvent energies of different graph matrices.