A spectral excess theorem for nonregular graphs

Guang Siang Lee*, Chih-wen Weng

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


The spectral excess theorem asserts that the average excess is, at most, the spectral excess in a regular graph, and equality holds if and only if the graph is distance-regular. An example demonstrates that this theorem cannot directly apply to nonregular graphs. This paper defines average weighted excess and generalized spectral excess as generalizations of average excess and spectral excess, respectively, in nonregular graphs, and proves that for any graph the average weighted excess is at most the generalized spectral excess. Aside from distance-regular graphs, additional graphs obtain the new equality. We show that a graph is distance-regular if and only if the new equality holds and the diameter D equals the spectral diameter d. For application, we demonstrate that a graph with odd-girth 2. d+. 1 must be distance-regular, generalizing a recent result of van Dam and Haemers.

Original languageEnglish
Pages (from-to)1427-1431
Number of pages5
JournalJournal of Combinatorial Theory. Series A
Issue number7
StatePublished - 1 Oct 2012


  • Distance-regular graphs
  • Eigenvalues
  • Spectral excess theorem


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