ABSTRACT
Let G be a connected graph with distance matrix , and let
be the diagonal matrix of vertex transmissions of G. For any
, the
-matrix of G is defined as
In this paper, we study the
-spectra of graphs. Firstly, the
-eigenvalues of some special graphs are presented. Then we give a lower bound on the kth smallest
-eigenvalue of graphs, and the extremal graphs are characterized. Also, several graph transformations on the
-spectral radius are given, as applications, some extremal graphs with given structure parameters are characterized. Finally, we give some properties when two graphs have the same
-spectra and several graphs are proved to be determined by their
-spectra.
KEYWORDS:
AMS CLASSIFICATION:
Acknowledgments
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.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1. If H is an invertible matrix, and x and y are two n-dimensional column vectors, then