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arXiv · 1901.07695

On the generalized distance spectral radius of graphs

Shu-Yu Cui·Gui-Xian Tian·Lu Zheng
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Abstract

The generalized distance spectral radius of a connected graph $G$ is the spectral radius of the generalized distance matrix of $G$, defined by $$D_\alpha(G)=\alpha Tr(G)+(1-\alpha)D(G), \;\;0\le\alpha \le 1,$$ where $D(G)$ and $Tr(G)$ denote the distance matrix and diagonal matrix of the vertex transmissions of $G$, respectively. This paper characterizes the unique graph with minimum generalized distance spectral radius among the connected graphs with fixed chromatic number, which answers a question about the generalized distance spectral radius in spectral extremal theories. In addition, we also determine graphs with minimum generalized distance spectral radius among the $n$-vertex trees and unicyclic graphs, respectively. These results generalize some known results about distance spectral radius and distance signless Laplacian spectral radius of graphs.

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Shu-Yu Cui, Gui-Xian Tian, Lu Zheng. 2019-01-23. On the generalized distance spectral radius of graphs. https://arxiv.org/abs/1901.07695

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