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Strongly Regular Square-free Graphs with μ=2

✍ Scribed by Benjamin V.C. Collins

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
277 KB
Volume
18
Category
Article
ISSN
0195-6698

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✦ Synopsis

By a square in an undirected graph ⌫ , we mean a cycle x , y , z , w such that x is not adjacent to z and y is not adjacent to w . Suppose that ⌫ is a strongly regular graph with ϭ 2 , and assume that ⌫ does not contain a square . Pick any vertex x of ⌫ and let ⌫ Ј denote the induced subgraph on the first subsconstituent of ⌫ with respect to x . It is known that ⌫ Ј is strongly regular . We express the parameters of ⌫ and ⌫ Ј in terms of two integer variables , p and h , which do not depend on x . We present feasibility conditions which must be satisfied by p and h , and show that , if p satisfies these , then p ϭ ( f 2 i ϩ 1 ) 2 for some integer i (0 р i Ͻ ϱ ) , where f 0 , f 1 , f 2 , . . . is the Fibonacci sequence , defined by f 0 ϭ 0 , f 1 ϭ 1 and f j ϭ f j Ϫ 1 ϩ f j Ϫ 2 (2 р j р ϱ ) .

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