Classification of amply regular graphs with \(b_1=6\). (English. Russian original) Zbl 1304.05033

Efimov, K. S.

Proc. Steklov Inst. Math. 283, Suppl. 1, S46-S55 (2013); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 18, No. 3, 90-98 (2012).

Summary: An undirected graph with \(v\) vertices in which the degrees of all vertices are equal to \(k\), each edge is contained in exactly \(\lambda\) triangles, and the intersection of the neighborhoods of any two vertices at distance 2 contains exactly \(\mu\) vertices is called amply regular with parameters (\(v\), \(k\), \(\lambda\), \(\mu\)). We complete the classification of amply regular graphs with \(b_1=6\), where \(b_1=k-\lambda -1\).

MSC:

05C12	Distance in graphs
05C07	Vertex degrees

Keywords:

amply regular graph; distance-regular graph

References:

[1]	A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer-Verlag, Berlin, 1989). · Zbl 0747.05073
[2]	A. A. Makhnev, ”On the strong regularity of some edge-regular graphs,” Izv. Math. 68(1), 159–180 (2004). · Zbl 1080.05100
[3]	S. A. Vasil’ev and A. A. Makhnev, ”On amply regular graphs with b 1 = 4,” Izv. Gomel’sk. Gos. Univ. 3, 101–108 (2006).
[4]	V. I. Kazarina and A. A. Makhnev, ”On edge-regular graphs with b 1 = 5,” Vladikavkaz. Mat. Zh. 11(1), 29–42 (2009). · Zbl 1324.05159
[5]	K. S. Efimov and A. A. Makhnev, ”Amply regular graphs with b 1 = 6,” Zh. Sib. Fed. Univ. 2(1), 63–77 (2009).
[6]	K. S. Efimov, A. A. Makhnev, and M. S. Nirova, ”On amply regular graphs with k = 10 and {\(\lambda\)} = 3,” Trudy Inst. Mat. Mekh. UrO RAN 16(2), 75–90 (2010).
[7]	K. S. Efimov and A. A. Makhnev, ”On amply regular graphs with k = 11 and {\(\lambda\)} = 4,” in Proc. Internat. Conf. on Algebra and Geometry (Yekaterinburg, 2011), pp. 75–90 [in Russian].
[8]	A. A. Makhnev, ”On extensions of partial geometries containing small {\(\mu\)}-subgraphs,” Diskret. Anal. Issled. Oper. 3(3), 71–83 (1996). · Zbl 0934.51005