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On edge-regular graphs with \(k\geq 3b_1-3\). (English. Russian original) Zbl 1157.05338
St. Petersbg. Math. J. 18, No. 4, 517-538 (2007); translation from Algebra Anal. 18, No. 4, 10-38 (2006).
Summary: An undirected graph on \( v\) vertices in which the degrees of all vertices are equal to \( k\) and each edge belongs to exactly \( \lambda\) triangles is said to be edge-regular with parameters \( (v,k,\lambda)\). It is proved that an edge-regular graph with parameters \( (v,k,\lambda)\) such that \( k\geq 3b_1-3\) either has diameter 2 and coincides with the graph \( P(2)\) on 20 vertices or with the graph \( M(19)\) on 19 vertices; or has at most \( 2k+4\) vertices; or has diameter at least 3 and is a trivalent graph without triangles, or the line graph of a quadrivalent graph without triangles, or a locally hexagonal graph; or has diameter 3 and satisfies \( |\Gamma_3(u)|\leq 1\) for each vertex \( u\).
MSC:
05E30 Association schemes, strongly regular graphs
05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
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