On the Geršgorin discs of distance matrices of graphs

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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 (2018) 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 disc? In this paper, we provide a negative answer by constructing an infinite family of counterexamples.

, 11 pages

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The Electronic Journal of Linear Algebra, 37, 709–717, 2021 BibTeX reference