The sum of the largest and smallest signless laplacian eigenvalues and some Hamiltonian properties of graphs

Abstract

The signless Laplacian eigenvalues of a graph $G$ are eigenvalues of the matrix $Q(G) = D(G) + A(G)$, where $D(G)$ is the diagonal matrix of the degrees of the vertices in $G$ and $A(G)$ is the adjacency matrix of $G$. Using a result on the sum of the largest and smallest signless Laplacian eigenvalues obtained by Das in \cite{Das}, we in this note present sufficient conditions based on the sum of the largest and smallest signless Laplacian eigenvalues for some Hamiltonian properties of graphs.