Iofinova and Ivanov (1985) showed that there exist exactly five bipartitecubic semisymmetric graphs whose automorphism
groups preserves the bipartite parts and acts primitively on each part. These graphs
have 110, 126, 182, 506, and 990 vertices, and their automorphism groups are , , , , and , respectively, where is one of the Mathieu groups.
The smallest Iofinova-Ivanov graph is the graph on 110 vertices illustrated above in two embeddings which is the second smallest cubic
semisymmetric graph (Iofinova and Ivanov 2002, Marušič et al.
2005). It was constructed in Ivanov (1983) in terms of the Paley design .
The 110-vertex Iofinova-Ivanov graph is illustrated above in three LCF
notations of degree 11 and four of degree 5.
Biggs, N. L. Algebraic Graph Theory. Cambridge, England: Cambridge University Press, p. 164,
1974.Chuvaeva, I. V. "On Some Combinatoria Objects which Admit
the Mathieu Group ." In Methods for Complex System Studies.
Moscow: VNIISI, pp. 47-52, 1983.Conder, M.; Malnič, A.; Marušič,
D.; Pisanski, T.; and Potočnik, P. "The Ljubljana Graph." 2002. http://citeseer.ist.psu.edu/conder02ljubljana.html.Faradžev,
I. A.; Klin, M. H.; and Muzichuk, M. E. "Cellular Rings and Groups
of Automorphisms of Graphs."Iofinova, M. E. and Ivanov, A. A.
"Bi-Primitive Cubic Graphs." In Investigations in the Algebraic Theory
of Combinatorial Objects. pp. 123-134, 2002. (Vsesoyuz. Nauchno-Issled.
Inst. Sistem. Issled., Moscow, pp. 137-152, 1985.)Ivanov, A. A.
"Computation of Lengths of Orbits of a Subgroup in a Transitive Permutation
Group." In Methods for Complex System Studies. Moscow: VNIISI, pp. 3-7,
1983.Ivanov, A. V. "On Edge But Not Vertex Transitive Regular
Graphs." In Combinatorial
Design Theory (Ed. C. J. Colbourn and R. Mathon). Amsterdam,
Netherlands: North-Holland, pp. 273-285, 1987.Marušič,
D.; Pisanski, T.; and Wilson, S. "The Genus of the Gray Graph is 7." Europ.
J. Combin.26, 377-385, 2005.Ronan, M. A. and Stroth,
G. "Minimal Parabolic Geometries of the Sporadic Groups." Europ. J.
Combin.5, 59-91, 1984.Wong, W. "Determination of a
Class of Primitive Permutation Groups." Math. Z.99, 235-246,
1967.
, 
, 
, 
, and 
, respectively, where 
is one of the Mathieu groups.
.
for 
and 23, respectively (Iofinova and Ivanov 2002).
.
." In Methods for Complex System Studies.
Moscow: VNIISI, pp. 47-52, 1983.Conder, M.; Malnič, A.; Marušič,
D.; Pisanski, T.; and Potočnik, P. "The Ljubljana Graph." 2002.