Non- primitive strongly regular graph of the suffi -

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Non-primitive strongly regular graph of sufficient conditions

zhou, associateprofessorofQinghalnati0nalitiesilIstitute. A - ■ lI bad floating down what _2 1i Fuyang Teachers College (Natural Science) Volume No. lg (k, a k a 1) = 1then five Ic. Sincec ≤ ≤ k, thisimpliesthatf == kofc = O, hencethatGisimprimitive · Weknown2copiesofK5isimprimitive, thisbecause (Chi, a 1) = (4,9) = 1. Butitcan'tuselemma3, becauseP a 9isnotprime. Clearlytheconditionoftheorem1weakerthan. thecondition0f, lemma3 · CoroIlaryIfGisastronglyregulargraphwithkandkcoprime, thenGisimprimitive · ProofFrom (1) wehave, (Chi, Chi) = (Chi, a k-1) a 1 (Chi, a 1) a 1Bytheorem1, Gisimprimitive. Theorem2LetGbean (, chi, a, c)-stronglyregulargraph, ifa: May 1, thenGisimprimitive · Proofk (Chi-1 A a) A (a k a 1) cifn-k a 1, then (a k a 1) c of a 0 a k a 1-0OFf = 0when Bamboo One is a 1-0, since a port of a k 1, thisimpliesGK. Gisnotstronglyregulargraph, thiscontradictory. henceC 一 0, Gisimprimitive. Theore Ding 3An (, k, mouth, c)-stronglyregulargraphisimprimitiveifandonlyiff = five orc = O · ProofIfitisimprimitivethenc = Oorc a foot. Whenc-k, from (2) wehavea = k a 1, thenfrom (1) wehave one by one 2 Chi + a-n-1 a 2k + Chi Chi a 1 = WhenC eleven fifteen, from (2) wehave mouth of a 2 foot a bundle, 10. somereason, an O: c = Chi, a c 10. Itisimprimitive. CorollaryAn (n, a k, a a, a c)-stronglyregulargraphisimprimitiveifandonlyifc a 0orc = five References1J. J. Seide1, Stronglyregulargraphs, anintroduction, insurveysincombinatorics'Prom · 7-thBritishcombinatoricsconference, editedbyB. BoLLobas,[link widoczny dla zalogowanych], LondonMath. Soe. LectureNoteSeries38, Combridge1979, pp157 ~ 180 ·. 2Godsil. C. D. A1gebraiccombinatorics, Firstpublishedin1993bychapmanandHall · NewYork non-primitive strongly regular graph of the sufficient conditions for HU students Biao (Qinghai Institute of Applied Mathematics, Qinghai, Xining 810007) Abstract Let G be a (n,, n, c) a strongly regular graph, (, t,, A a,) is its complement. If their parameters satisfy the following conditions; 1) V.,, l-1 relatively prime; 2), i coprime; 3) with a bulging 1, then G is non primitive. Non-primitive G if and only if c is a positive or c 10. Key words strongly regular graph; primitive; non-primitive

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