Distance signless Laplacian eigenvalues of graphs

Kinkar Chandra DAS; Huiqiu LIN; Jiming GUO

doi:10.1007/s11464-019-0779-3

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Front. Math. China ›› 2019, Vol. 14 ›› Issue (4) : 693-713. DOI: 10.1007/s11464-019-0779-3

RESEARCH ARTICLE

Distance signless Laplacian eigenvalues of graphs

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Abstract

Suppose that the vertex set of a graph G is $V (G) = {v 1, v 2, ..., v n}$ . The transmission $T r (v i)$ (or D_i) of vertex vi is defined to be the sum of distances from vi to all other vertices. Let $T r (G)$ be the $n × n$ diagonal matrix with its (i, i)-entry equal to $T r G (v i)$ . The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as $L (G) = T r (G) + D (G)$ , where $D (G)$ is the distance matrix of G. In this paper, we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible. We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs. Moreover, we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees, and characterize extremal graphs.

Keywords

Graph / distance signless Laplacian spectral radius / second largest eigenvalue of distance signless Laplacian matrix / spread

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Kinkar Chandra DAS, Huiqiu LIN, Jiming GUO. Distance signless Laplacian eigenvalues of graphs. Front. Math. China, 2019, 14(4): 693‒713 https://doi.org/10.1007/s11464-019-0779-3

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