New results on the Dα-matrix of connected graphs

Let G be a simple undirected connected graph. Let D(G) be the distance matrix of G and let Tr(G) be the diagonal matrix of the vertex transmissions in G. Let α∈[0,1]. In S-Y. Cui et al. (2019) [7] the matrixDα(G)=αTr(G)+(1−α)D(G) is introduced and several properties are obtained. In this paper, new...

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Bibliographic Details
Published in:Linear algebra and its applications 2019-09, Vol.577, p.168-185
Main Authors: Díaz, Roberto C., Pastén, Germain, Rojo, Oscar
Format: Article
Language:English
Subjects:
Apexes
Cluster
Distance matrix
Distance signless Laplacian matrix
Graph theory
Graphs
H-join
Inequalities
Linear algebra
Multipartite graph
Spectral radius
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Summary:Let G be a simple undirected connected graph. Let D(G) be the distance matrix of G and let Tr(G) be the diagonal matrix of the vertex transmissions in G. Let α∈[0,1]. In S-Y. Cui et al. (2019) [7] the matrixDα(G)=αTr(G)+(1−α)D(G) is introduced and several properties are obtained. In this paper, new properties on the Dα-matrix are derived including inequalities that involve the largest vertex transmission and the spectral radii of the distance matrix, distance signless Laplacian matrix and Dα-matrix. The necessary and sufficient condition for the equality in each of the inequalities is given. Moreover, some results on the Dα-matrix of a graph with independent sets of vertices sharing the same set of neighbors, including the case of a complete multipartite graph, are obtained. Finally, the spectrum of Dα(G) is determined when G is the H-join of regular graphs.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2019.04.030