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.
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References
- [1] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan, London and Elsevier, New York (1976).
- [2] K. C. Das, Proof of conjectures involving the largest and the smallest signless Laplacian eigenvalues of graphs, Discrete Mathematics 312 (2012) 992 – 998.
