The Signless Laplacian Estrada Index of Unicyclic Graphs

For a graph $G$, the signless Laplacian Estrada index is defined as $SLEE(G)={\sum }_{i=1}^n{e}^{q_i^{}}$,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 $SLEE$ and then determine the unique unicyclic graph with maximum $SLEE$ among the unicyclic graphs on $n$ vertices with given diameter.