Designs, Codes and Cryptography

, Volume 34, Issue 2–3, pp 241–248

Algebraic Characterizations of Graph Regularity Conditions

Article

Abstract

It is well-known that a connected finite simple graph is regular if and only if the all-ones matrix spans an ideal of its adjacency algebra. We show that several other graph regularity conditions involving pairs and triples of vertices also have ideal theoretic characterizations in some appropriate algebras.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bannai, E., Ito, T. 1984Algebraic Combinatorics IBenjamin/CummingsMenlo ParkMATHGoogle Scholar
  2. Biggs, N. 1993Algebraic Graph TheoryCambridge University PressCambridge2nd ed.Google Scholar
  3. Brouwer, A. E., Neumaier, A. 1989Distance-Regular GraphsSpringerNew YorkMATHGoogle Scholar
  4. Collins, B. V. C. 1997The girth of a thin distance-regular graphGraphs Combin132130MATHMathSciNetGoogle Scholar
  5. Curtin, B. 2000Almost 2-homogeneous bipartite distance-regular graphsEur. J. Combinatorics21865876MATHMathSciNetGoogle Scholar
  6. B. Curtin and K. Nomura, 1-Homogeneous, pseudo 1-homogenous, and 1-thin distance-regular graphs J. Combin. Theory Ser. B., to appear.Google Scholar
  7. Dammerell, R. M. 1973On Moore graphsProc. Cambridge Philos. Soc.74227236CrossRefMathSciNetGoogle Scholar
  8. C. Delorme, Régularité métrique forte, Rapport de Rechere No. 156 Univ. Pais Sud, Orsay (1983).Google Scholar
  9. Fiol, M. A., Garriga, E., Yebra, J. L. A. 1996Locally pseudo-distance-regular graphsJ. Combin. Theory Ser. B68179205MATHMathSciNetGoogle Scholar
  10. Fiol, M. A., Garriga, E. 1999On the algebraic theory of pseudo-distance-regularity around a setLinear Algebra Appl.298115141MATHMathSciNetGoogle Scholar
  11. Fiol, M. A. 2002Algebraic characterizations of distance-regular graphsDiscrete Math.2461111129MathSciNetGoogle Scholar
  12. Godsil, C. D., Shaw-Taylor, J. 1987Distance-regularized graphs are distance-regular or distance-biregularJ. Combin. Theory Ser. B431424MATHMathSciNetGoogle Scholar
  13. Hoffman, A. J. 1963On the polynomial of a graphAm. Math. Monthly703036MATHGoogle Scholar
  14. Jaeger, F. 1995On spin models, triply regular association schemes, and dualityJ. Alg. Combin.4103144MATHMathSciNetGoogle Scholar
  15. Nomura, K. 1994Homogeneous graphs and regular near polygonsJ. Combin. Theory Ser. B606371MATHMathSciNetGoogle Scholar
  16. Smith, M. S. 1975On rank 3 permutation groupsJ. Algebra332242MATHMathSciNetGoogle Scholar
  17. P. Terwilliger, The subconstituent algebra of an association scheme, J. Alg. Combin. (Part I) Vol. 1 (1992) pp. 363–388; (Part II) Vol. 2 (1993) pp. 73--103; (Part III) Vol. 2 (1993) pp. 177--210.Google Scholar
  18. Weichsel, P. M. 1982On distance-regularity in graphsJ. Combin. Theory Ser. B32156161MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of South FloridaTampaUSA

Personalised recommendations