The signless Laplacian Estrada index of tricyclic graphs

Abstract

‎For a simple graph $G$‎, ‎the signless Laplacian Estrada index is defined as $SLEE(G)=sum^{n}_{i=1}e^{q^{}_i}$‎, ‎where $q^{}_1‎, ‎q^{}_2‎, ‎dots‎, ‎q^{}_n$ 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.