On the second largest distance eigenvalue of a graph

Abstract

Let G be a simple connected graph of order n and D(G) be the distance matrix of G. Suppose that is the distance spectrum of G. A graph G is said to be determined by its D-spectrum if any graph with the same distance spectrum as G is isomorphic to G. In this paper, we consider spectral characterization on the second largest distance eigenvalue of graphs, and prove that the graphs with are determined by their D-spectra.

Keywords:

• Distance matrix
• the second largest distance eigenvalue
• D-spectrum determined

AMS Subject Classifications:

• 05C50
• 05C12

Acknowledgements

The authors would like to thank the anonymous referees very much for valuable suggestions and corrections which lead to a great improvement in the original paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Ruifang Liu was supported by NSFC [grant number 11201432], [grant number 11571323] and Outstanding Young Talent Research Fund of Zhengzhou University [grant number 1521315002]. Litao Guo was supported by NSFC [grant number 11301440] and NSF-Fujian [grant number JA13240].