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Algebraic Characterizations of Graph Regularity Conditions

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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.

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References

  • E. Bannai T. Ito (1984) Algebraic Combinatorics I Benjamin/Cummings Menlo Park Occurrence Handle0555.05019

    MATH  Google Scholar 

  • N. Biggs (1993) Algebraic Graph Theory Cambridge University Press Cambridge

    Google Scholar 

  • A. E. Brouwer A. Neumaier (1989) Distance-Regular Graphs Springer New York Occurrence Handle0747.05073

    MATH  Google Scholar 

  • B. V. C. Collins (1997) ArticleTitleThe girth of a thin distance-regular graph Graphs Combin 13 21–30 Occurrence Handle0878.05084 Occurrence Handle1435654

    MATH  MathSciNet  Google Scholar 

  • B. Curtin (2000) ArticleTitleAlmost 2-homogeneous bipartite distance-regular graphs Eur. J. Combinatorics 21 865–876 Occurrence Handle1002.05069 Occurrence Handle1787900

    MATH  MathSciNet  Google Scholar 

  • B. Curtin and K. Nomura, 1-Homogeneous, pseudo 1-homogenous, and 1-thin distance-regular graphs J. Combin. Theory Ser. B., to appear.

  • R. M. Dammerell (1973) ArticleTitleOn Moore graphs Proc. Cambridge Philos. Soc. 74 227–236 Occurrence Handle10.1017/S0305004100048015 Occurrence Handle318004

    Article  MathSciNet  Google Scholar 

  • C. Delorme, Régularité métrique forte, Rapport de Rechere No. 156 Univ. Pais Sud, Orsay (1983).

  • M. A. Fiol E. Garriga J. L. A. Yebra (1996) ArticleTitleLocally pseudo-distance-regular graphs J. Combin. Theory Ser. B 68 179–205 Occurrence Handle0861.05064 Occurrence Handle1417795

    MATH  MathSciNet  Google Scholar 

  • M. A. Fiol E. Garriga (1999) ArticleTitleOn the algebraic theory of pseudo-distance-regularity around a set Linear Algebra Appl. 298 115–141 Occurrence Handle0984.05060 Occurrence Handle1723119

    MATH  MathSciNet  Google Scholar 

  • M. A. Fiol (2002) ArticleTitleAlgebraic characterizations of distance-regular graphs Discrete Math. 246 1111–129 Occurrence Handle1887483

    MathSciNet  Google Scholar 

  • C. D. Godsil J. Shaw-Taylor (1987) ArticleTitleDistance-regularized graphs are distance-regular or distance-biregular J. Combin. Theory Ser. B 43 14–24 Occurrence Handle0616.05041 Occurrence Handle897237

    MATH  MathSciNet  Google Scholar 

  • A. J. Hoffman (1963) ArticleTitleOn the polynomial of a graph Am. Math. Monthly 70 30–36 Occurrence Handle0112.14901

    MATH  Google Scholar 

  • F. Jaeger (1995) ArticleTitleOn spin models, triply regular association schemes, and duality J. Alg. Combin. 4 103–144 Occurrence Handle0820.05063 Occurrence Handle1323744

    MATH  MathSciNet  Google Scholar 

  • K. Nomura (1994) ArticleTitleHomogeneous graphs and regular near polygons J. Combin. Theory Ser. B 60 63–71 Occurrence Handle0793.05130 Occurrence Handle1256584

    MATH  MathSciNet  Google Scholar 

  • M. S. Smith (1975) ArticleTitleOn rank 3 permutation groups J. Algebra 33 22–42 Occurrence Handle0297.20002 Occurrence Handle384905

    MATH  MathSciNet  Google Scholar 

  • 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.

  • P. M. Weichsel (1982) ArticleTitleOn distance-regularity in graphs J. Combin. Theory Ser. B 32 156–161 Occurrence Handle0477.05047 Occurrence Handle657684

    MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, University of South Florida, 4202 E. Fowler Ave. PHY114, Tampa, FL, 33620, USA

    Brian Curtin

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  1. Brian Curtin
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Correspondence to Brian Curtin.

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Curtin, B. Algebraic Characterizations of Graph Regularity Conditions. Des Codes Crypt 34, 241–248 (2005). https://doi.org/10.1007/s10623-004-4857-4

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