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arXiv · 1705.07419

More results on the distance (signless) Laplacian eigenvalues of graphs

Jie Xue·Huiqiu Lin·Kinkar Ch. Das·Jinlong Shu
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Abstract

Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. Let $Tr(G)$ be the diagonal matrix of vertex transmissions of $G$ and $D(G)$ be the distance matrix of $G$. The distance Laplacian matrix of $G$ is defined as $\mathcal{L}(G)=Tr(G)-D(G)$. The distance signless Laplacian matrix of $G$ is defined as $\mathcal{Q}(G)=Tr(G)+D(G)$. In this paper, we give a lower bound on the distance Laplacian spectral radius in terms of $D_1$, as a consequence, we show that $\partial_1^L(G)\geq n+\lceil\frac{n}{\omega}\rceil$ where $\omega$ is the clique number of $G$. Furthermore, we give some graft transformations, by using them, we characterize the extremal graph attains the maximum distance spectral radius in terms of $n$ and $\omega$. Moreover, we also give bounds on the distance signless Laplacian eigenvalues of $G$, and give a confirmation on a conjecture due to Aouchiche and Hansen.

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Jie Xue, Huiqiu Lin, Kinkar Ch. Das, Jinlong Shu. 2017-05-21. More results on the distance (signless) Laplacian eigenvalues of graphs. https://arxiv.org/abs/1705.07419

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