Geodesic
On edge-regular graphs with $k\ge 3b_1-3$
Algebra i analiz, Tome 18 (2006) no. 4, pp. 10-38.

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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 $k\ge 3b_1-3$ 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 $|\Gamma_3(u)|\le 1$ for each vertex $u$. 
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I. N. Belousov; A. A. Makhnev. On edge-regular graphs with $k\ge 3b_1-3$. Algebra i analiz, Tome 18 (2006) no. 4, pp. 10-38. http://geodesic.mathdoc.fr/item/AA_2006_18_4_a1/

[1] Brouwer A. E., Cohen A. M., Neumaier A., Distance-regular graphs, Ergeb. Math. Grenzgeb. (3), 18, Springer-Verlag, Berlin etc., 1989 | MR | Zbl

[2] Makhnev A. A., Minakova I. M., “Ob odnom klasse reberno regulyarnykh grafov”, Izv. Gomelsk. gos. un-ta, Voprosy algebry, 3 (2000), 145–154 | Zbl

[3] Zaripov S. R., Makhnev A. A., Yablonko I. P., “Reberno regulyarnye grafy diametra $\lambda\ge 2k/3-2$”, Tr. Ukrain. mat. kongressa, sektsiya 1: Algebra i teoriya chisel (Kiev, 2001), In-t mat. NAN Ukrainy, Kiev, 2003, 46–61 | MR

[4] Makhnev A. A., “O silnoi regulyarnosti nekotorykh reberno regulyarnykh grafov”, Izv. RAN. Ser. mat., 68:1 (2004), 159–182 | MR

[5] Makhnev A. A., Vedenev A. A., Kuznetsov A. N., Nosov V. V., “O khoroshikh parakh v reberno regulyarnykh grafakh”, Diskret. matem., 15:1 (2003), 77–97 | MR | Zbl

[6] Belousov I. N., Gurskii E. I., Dergach A. S., Makhnev A. A., “O pochti khoroshikh parakh v reberno regulyarnykh grafakh”, Probl. teor. i prikl. mat., Tr. molodezh. konf., Ekaterinburg, 2004, 9–11

[7] Kabanov V. V., Makhnev A. A., “Ob otdelimykh grafakh s nekotorymi usloviyami regulyarnosti”, Matem. sb., 187:10 (1996), 73–86 | MR | Zbl

[8] Belousov I. N., Makhnev A. A., “O pochti khoroshikh parakh vershin v reberno regulyarnykh grafakh”, Izv. Ural. gos. un-ta. Mat. mekh., 2005, no. 7(36), 35–48 | MR