Maximizing the signless Laplacian spectral radius of graphs with given diameter or cut vertices

Abstract

The signless Laplacian matrix of a graph is defined to be the sum of its adjacency matrix and degree matrix. Let be the set of all the connected graphs of order n and diameter d and 𝒢 _n,k the set of all connected graphs with order n and k cut vertices. In this article, we determine the graphs that have the maximal signless Laplacian spectral radius and give the upper bounds of graphs in these two sets.

Keywords:

• adjacency matrix
• signless Laplacian
• diameter
• cut vertices
• spectral radius

AMS Subject Classification:

• 05C50

Acknowledgements

The authors are grateful to Prof Slobodan K. Simić and Dr Mingqing Zhai for sending their papers 16 and 20, respectively. The authors would also like to thank the anonymous referee for his or her many valuable suggestions towards improving this article. Supported by the National Science Foundation of China (No. 10961023) and by the Scientific Research Innovation Program of Qinghai Normal University.