Open AccessOn the Ger\v{s}gorin disks of distance matrices of graphs
Author(s) -
Mustapha Aouchiche,
Bilal A. Rather,
Issmail El Hallaoui
Publication year - 2021
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/ela.2021.6489
Subject(s) - mathematics , counterexample , combinatorics , spectral radius , distance matrix , laplacian matrix , eigenvalues and eigenvectors , vertex (graph theory) , laplace operator , resistance distance , matrix (chemical analysis) , diagonal , graph , discrete mathematics , graph power , line graph , physics , mathematical analysis , geometry , chemistry , quantum mechanics , chromatography
For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)=Tr(G)-D(G)$, and $D^{Q}(G)=Tr(G)+D(G)$ be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix, and the distance signless Laplacian matrix of $G$, respectively. Atik and Panigrahi [2] suggested the study of the problem: Whether all eigenvalues, except the spectral radius, of $ D(G) $ and $ D^{Q}(G) $ lie in the smallest Ger\v{s}gorin disk? In this paper, we provide a negative answer by constructing an infinite family of counterexamples.