‎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 $SLEE$ a...

SLEE has various applications in a large variety of problems. The signless Laplacian Estrada index hypergraph H is defined as SLEE(H)=∑i=1neλi(Q), where λ1(Q),λ2(Q),…,λn(Q) are the eigenvalues matrix H. In this paper, we characterize unique r-uniform unicyclic hypergraphs with maximum and minimum SLEE.

In this paper, we present a sharp upper and lower bounds for the signless Laplacian spectral radius of graphs in terms of clique number. Moreover, the extremal graphs which attain the upper and lower bounds are characterized. In addition, these results disprove the two conjectures on the signless Laplacian spectral radius in [P. Hansen and C. Lucas, Bounds and conjectures for the signless Lapla...

A signless Laplacian eigenvalue of a graph G is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, some necessary and sufficient conditions for a graph with one main signless Laplacian eigenvalue or two main signless Laplacian eigenvalues are given. And the trees and unicyclic graphs with exactly two main signless L...

Abstract The Laplacian Estrada index of a graphG is defined as LEE(G) = ∑n i=1 e μi , where μ1 ≥ μ2 ≥ · · · ≥ μn−1 ≥ μn = 0 are the eigenvalues of its Laplacian matrix. An unsolved problem in [19] is whether Sn(3, n − 3) or Cn(n − 5) has the third maximal Laplacian Estrada index among all trees on n vertices, where Sn(3, n − 3) is the double tree formed by adding an edge between the centers of ...

let $g = (v, e)$ be a simple graph. denote by $d(g)$ the diagonal matrix $diag(d_1,cdots,d_n)$, where $d_i$ is the degree of vertex $i$  and  $a(g)$ the adjacency matrix of $g$. the  signless laplacianmatrix of $g$ is $q(g) = d(g) + a(g)$ and the $k-$th signless laplacian spectral moment of  graph $g$ is defined as $t_k(g)=sum_{i=1}^{n}q_i^{k}$, $kgeqslant 0$, where $q_1$,$q_2$, $cdots$, $q_n$ ...