On spectral spread of generalized distance matrix of a graph

ABSTRACT

For a simple connected graph G, let $D\left(G\right)$, $Tr\left(G\right)$, $D^L\left(G\right)$ and $D^Q\left(G\right)$, respectively, are the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix. The generalized distance matrix $D_{\alpha }\left(G\right)$ of G is the convex linear combinations of $Tr\left(G\right)$ and $D\left(G\right)$ and is defined as $D_{\alpha }\left(G\right)=\alpha Tr\left(G\right)+(1-\alpha )D\left(G\right)$, for $0\le \alpha \le 1$. As $D_0\left(G\right)=D\left(G\right),2D_{\frac{1}{2}}\left(G\right)=D^Q\left(G\right),D_1\left(G\right)=Tr\left(G\right)$ and $D_{\alpha }\left(G\right)-D_{\text{β}}\left(G\right)=(\alpha -\text{β})D^L\left(G\right)$, this matrix reduces to merging the distance spectral and distance signless Laplacian spectral theories. Let ${\mathrm{\partial }}_1\left(G\right)\ge {\mathrm{\partial }}_2\left(G\right)\ge \cdots \ge {\mathrm{\partial }}_n\left(G\right)$ be the eigenvalues of $D_{\alpha }\left(G\right)$ and let $S_{D_{\alpha }}\left(G\right)={\mathrm{\partial }}_1\left(G\right)-{\mathrm{\partial }}_n\left(G\right)$ be the generalized distance spectral spread of the graph G. In this paper, we obtain bounds for the generalized distance spectral spread $S_{D_{\alpha }}\left(G\right)$. We also obtain a relation between the generalized distance spectral spread $S_{D_{\alpha }}\left(G\right)$ and the distance spectral spread $S_D\left(G\right)$. Further, we obtain lower bounds for $S_{D_{\alpha }}\left(G\right)$ of bipartite graphs involving different graph parameters and we characterize the extremal graphs for some cases. We also obtain lower bounds for $S_{D_{\alpha }}\left(G\right)$ in terms of clique number and independence number of the graph G and characterize the extremal graphs for some cases.

Keywords:

• Distance spectrum (matrix)
• distance signless Laplacian spectrum (matrix)
• transmission regular graph
• generalized distance matrix
• generalized distance spectral spread

AMS Subject Classifications:

• Primary: 05C50
• 05C12
• Secondary: 15A18

Acknowledgments

We sincerely thank the three anonymous referees whose valuable comments and suggestions resulted in the improvement to the presentation of this paper. The research of S. Pirzada is supported by SERB-DST, New Delhi under the research project number MTR/2017/000084.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of S. Pirzada is supported by SERB-DST, New Delhi under the research project number MTR/2017/000084.