Permanental bounds for the signless Laplacian matrix of bipartite graphs and unicyclic graphs

Abstract

Let G be a graph and let Q(G) be its signless Laplacian matrix. When G is a unicyclic (respectively, bipartite) graph we obtain sharp upper and lower bounds for the permanent of Q(G) in terms of the order of G. Improved bounds are obtained in terms of the given girth of G. In each of these cases, we characterize the extremal graphs.

Keywords:

• permanent
• signless Laplacian
• unicyclic graph
• girth

AMS Subject Classifications:

• 05C50
• 05C05
• 15A15

Acknowledgements

The authors would like to express their sincere gratitude to the referees for a very careful reading of this article and for all their insightful comments and valuable suggestions, which led to a number of improvements in this article. The research is partially supported by self-determined research funds of CCNU (CCNU09Y01005, CCNU09Y01018) from the colleges' basic research and operation of MOE, and the National Science Foundation of China (Grant No. 11071096).