Sharp Bounds on the Signless Laplacian Estrada Index of Graphs

Shan Gao, Huiqing Liu

Let $G$ be a connected graph with $n$ vertices and $m$ edges. Let $q_1,q_2,\ldots,q_n$ be the eigenvalues of the signless Laplacian matrix of $G$, where $q_1\geq q_2\geq\cdots\geq q_n$. The signless Laplacian Estrada index of $G$ is defined as $\operatorname{SLEE}(G)=\sum^n_{i=1}e^{q_i}$. In this paper, we present some sharp lower bounds for $\operatorname{SLEE}(G)$ in terms of the $k$-degree and the first Zagreb index, respectively.