On the D_α-spectra of graphs

ABSTRACT

Let G be a connected graph with distance matrix $D\left(G\right)$, and let $Tr\left(G\right)$ be the diagonal matrix of vertex transmissions of G. For any $\alpha \in [0,1]$, the $D_{\alpha }$-matrix of G is defined as $D_{\alpha }\left(G\right)=\alpha Tr\left(G\right)+(1-\alpha )D\left(G\right).$ In this paper, we study the $D_{\alpha }$-spectra of graphs. Firstly, the $D_{\alpha }$-eigenvalues of some special graphs are presented. Then we give a lower bound on the kth smallest $D_{\alpha }$-eigenvalue of graphs, and the extremal graphs are characterized. Also, several graph transformations on the $D_{\alpha }$-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 $D_{\alpha }$-spectra and several graphs are proved to be determined by their $D_{\alpha }$-spectra.

KEYWORDS:

• $D_{\alpha }$-eigenvalue
• extremal graph
• cospectrality

AMS CLASSIFICATION:

• 05C50

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 $n\times n$ matrix, and x and y are two n-dimensional column vectors, then $det(H+x{y}^t)=det\left(H\right)(1+y^t{H}^{-1}x)$

Additional information

Funding

The first author was supported by the National Natural Science Foundation of China [grant number 11401211] and Fundamental Research Funds for the Central Universities [grant number 222201714049]. The third author was supported by the National Natural Science Foundation of China [grant number 11471121].