The signless Laplacian Estrada index of tricyclic graphs

Abstract

‎For a simple graph G‎, ‎the signless Laplacian Estrada index is defined as SLEE(G)=∑ni=1eqi‎, ‎where q1‎, ‎q2‎,...‎, ‎qn are the eigenvalues of the signless Laplacian matrix of G‎. ‎In this paper‎, ‎we first characterize the unicyclic graphs with the first two largest and smallest SLEE's and then determine the unique unicyclic graph with maximum SLEE among all ‎unicyclic graphs on n vertices with a given diameter‎. ‎All extremal graphs‎, ‎which have been introduced in our results are also extremal with respect to the signless Laplacian ‎resolvent energy‎.