The generalized distance matrix

Let D(G) and Diag(Tr) denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph G, respectively. The distance signless Laplacian matrix of G is defined as DQ(G)=Diag(Tr)+D(G). Heretofore, the spectral properties of D(G) and DQ(G) have attracted much more...

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Bibliographic Details
Published in:Linear algebra and its applications 2019-02, Vol.563, p.1-23
Main Authors: Cui, Shu-Yu, He, Jing-Xiang, Tian, Gui-Xian
Format: Article
Language:English
Subjects:
Distance matrix
Distance signless Laplacian matrix
Graph operation
Linear algebra
Lower bounds
Spectra
Spectral radius
Spectrum
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Summary:Let D(G) and Diag(Tr) denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph G, respectively. The distance signless Laplacian matrix of G is defined as DQ(G)=Diag(Tr)+D(G). Heretofore, the spectral properties of D(G) and DQ(G) have attracted much more attention. In the present paper, we propose to study the convex combinations Dα(G) of Diag(Tr) and D(G), defined asDα(G)=αDiag(Tr)+(1−α)D(G),0≤α≤1. This study sheds new light on D(G) and DQ(G). Some spectral properties of Dα(G) are given and a few open problems are discussed. Furthermore, we take effort to obtain some upper and lower bounds of spectral radius of Dα(G). Finally, the generalized distance spectra of some graphs obtained by operations are also studied.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2018.10.014