On the strong regularity of some edge-regular graphs. (English. Russian original) Zbl 1080.05100

Makhnev, A. A.

Izv. Math. 68, No. 1, 159-180 (2004); translation from Izv. Ross. Akad. Nauk Ser. Mat. 68, No. 1, 159-182 (2004).

The author studies undirected regular graphs such that each edge belongs to exactly \(\lambda\) triangles, the so-called edge-regular graphs. Examples are strongly regular graphs. In the book of A. E. Brouwer, A. M. Cohen and A. Neumaier [Distance-regular graphs (Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Springer, Berlin etc.) (1989; Zbl 0747.05073)] it is proved that a connected edge-regular graph on \(v\) vertices and of degree \(k\) is strongly regular, if \(\lambda\geq k+{1\over 2}- \sqrt{2k+2}\).
In the frame of this theorem the author conjectures that a connected edge-regular graph with parameters \((v, k,\lambda)\) satisfying \(\lambda> k -\sqrt{2k + 1}\) is strongly regular except for a concrete list of graphs. The main result of the paper is that the conjecture becomes true under the stronger assumption that \(\lambda\geq k+ 1/2- \sqrt{2k+8}\).

Reviewer: Udo Ott (Braunschweig)

MSC:

05E30

Association schemes, strongly regular graphs

Keywords:

strongly regular graphs

Citations:

Zbl 0747.05073