On a version of the spectral excess theorem

On a version of the spectral excess theorem

Given a regular (connected) graph G=(X,E) with adjacency matrix A, d+1 distinct eigenvalues, and diameter D, we give a characterization  of when its distance matrix AD is a polynomial in A, in terms of the adjacency spectrum of G and the arithmetic (or harmonic) mean of the numbers of vertices at di...

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Main Authors: Fiol, Miquel Àngel; Departament de Matem\`atiques, Universitat Polit\' ecnica de Catalunya, Barcelona Graduate School of Mathematics, Catalonia, Spain (Author), Penjic, Safet; Andrej Maru\v{s}i\v{c} Institute, University of Primorska, Muzejski trg 2 6000 Koper, Slovenia (Author)
Format: EJournal Article
Published: GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2020-10-16.
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Summary:Given a regular (connected) graph G=(X,E) with adjacency matrix A, d+1 distinct eigenvalues, and diameter D, we give a characterization  of when its distance matrix AD is a polynomial in A, in terms of the adjacency spectrum of G and the arithmetic (or harmonic) mean of the numbers of vertices at distance at most D-1 from every vertex. The same result is proved for any graph by using its Laplacian matrix L and corresponding spectrum. When D=d we reobtain the spectral excess theorem characterizing distance-regular graphs.
Item Description:https://www.ejgta.org/index.php/ejgta/article/view/904

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