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Some Spectral Characterizations of Strongly Distance-Regular Graphs

Published online by Cambridge University Press:  24 May 2001

M. A. FIOL
M. A. FIOL
Affiliation:
Departament de Matemàtica Aplicada i Telemàtica, Universitat Politècnica de Catalunya, Jordi Girona 1-3, Mòdul C3, Campus Nord, 08034 Barcelona, Spain (e-mail: fiol@mat.upc.es)
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Abstract

A graph Γ with diameter d is strongly distance-regular if Γ is distance-regular and its distance-d graph Γd is strongly regular. Some known examples of such graphs are the connected strongly regular graphs, with distance-d graph Γd = Γ (the complement of Γ), and the antipodal distance-regular graphs. Here we study some spectral conditions for a (regular or distance-regular) graph to be strongly distance-regular. In particular, for the case d = 3 the following characterization is proved. A regular (connected) graph Γ, with distinct eigenvalues λ0 > λ1 > λ2 > λ3, is strongly distance-regular if and only if λ2 = −1, and Γ3 is k-regular with degree k satisfying an expression which depends only on the order and the different eigenvalues of Γ.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 10 , Issue 2 , March 2001 , pp. 127 - 135
Copyright
2001 Cambridge University Press

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