Bounds for Generalized Distance Spectral Radius and the Entries of the Principal Eigenvector

Authors

DOI:

https://doi.org/10.5556/j.tkjm.52.2021.3280

Keywords:

Distance matrix (spectrum); Distance signlees Laplacian matrix (spectrum); generalized distance matrix; spectral radius; transmission regular graph; Perron vector

Abstract

For a simple connected graph $G$, the convex linear combinations $D_{\alpha}(G)$ of \ $Tr(G)$ and $D(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. As $D_{0}(G)=D(G)$, $2D_{\frac{1}{2}}(G)=D^{Q}(G)$, $D_{1}(G)=Tr(G)$ and $D_{\alpha}(G)-D_{\beta}(G)=(\alpha-\beta)D^{L}(G)$, this matrix reduces to merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral properties of the generalized distance matrix $D_{\alpha}(G)$. We obtain some lower and upper bounds for the generalized distance spectral radius, involving different graph parameters and characterize the extremal graphs. Further, we obtain upper and lower bounds for the maximal and minimal entries of the $ p $-norm normalized Perron vector corresponding to spectral radius $ \partial(G) $ of the generalized distance matrix $D_{\alpha}(G)$ and characterize the extremal graphs.

Author Biographies

Abdollah Alhevaz, Faculty of Mathematical Sciences, Shahrood University of Technology, Iran

Professor

Gui-Xian Tian, Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, P.R. China

professor

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Published

2021-01-31

How to Cite

Ahmad, H., Alhevaz, A., Baghipur, . M., & Tian, G.-X. (2021). Bounds for Generalized Distance Spectral Radius and the Entries of the Principal Eigenvector. Tamkang Journal of Mathematics, 52(1), 69-89. https://doi.org/10.5556/j.tkjm.52.2021.3280