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The geometry of t-spreads in k-walk-regular graphs

✍ Scribed by C. Dalfó; M. A. Fiol; E. Garriga

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
117 KB
Volume
64
Category
Article
ISSN
0364-9024

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✦ Synopsis

Abstract

A graph is walk‐regular if the number of closed walks of length ℓ rooted at a given vertex is a constant through all the vertices for all ℓ. For a walk‐regular graph G with d+1 different eigenvalues and spectrally maximum diameter D=d, we study the geometry of its d‐spreads, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a simplex (or tetrahedron in a three‐dimensional case) and we compute its parameters. Moreover, the results are generalized to the case of k‐walk‐regular graphs, a family which includes both walk‐regular and distance‐regular graphs, and their t‐spreads or vertices at distance t from each other. © 2009 Wiley Periodicals, Inc. J Graph Theory 64:312–322, 2010

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