Some extremal properties of the resolvent energy, Estrada and resolvent Estrada indices of graphs

Let G be a simple connected graph on n vertices and λ1,λ2,…,λn be the eigenvalues of the adjacency matrix of G. Estrada and Higham proposed an invariant of a graph G based on Taylor series expansion of spectral moments EE(G,c)=∑k=0∞ckMk(G). For ck=1nk (resp. 1k!, 1(n−1)k), EE(G,c) is the Resolvent energy (resp. Estrada index, Resolvent Estrada index) of G. In [18,17], Gutman et al. conjectured the structure of the extremal members of some classes of graphs by the aid of computer on Resolvent energy and Resolvent Estrada index, respectively. In [1], L. Allem et al. confirmed the validity of some of these conjectures on Resolvent energy. In this paper, we continue to study these indices based on these conjectures.