The smallest Iofinova-Ivanov graph is the graph on 110 vertices illustrated above in two drawings 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 . It has graph genus 15
(Conder and Stokes 2019).
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,
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A.; Marušič, D.; Pisanski, T.; and Potočnik, P. "The Edge-Transitive
but Not Vertex-Transitive Cubic Graph on 112 Vertices." J. Graph Th.50,
25-42, 2005. https://doi.org/10.1002/jgt.20089.Conder,
M. and Stokes, K. "New Methods for Finding Minimum Genus Embeddings of Graphs
on Orientable and Non-Orientable Surfaces." Ars. Math. Contemp.17,
1-35, 2019.Faradžev, I. A.; Klin, M. H.; and Muzichuk,
M. E. "Cellular Rings and Groups of Automorphisms of Graphs." In Investigations
in Algebraic Theory of Combinatorial Objects. (Ed. I. A. Faradžev,
A. A. Ivanov, M. H. Klin, and A. J. Woldar). Dordrecht,
Netherlands: Kluwer, pp. 1-152, 1994.Iofinova, M. E. and Ivanov,
A. A. "Bi-Primitive Cubic Graphs." In Investigations
in Algebraic Theory of Combinatorial Objects. (Ed. I. A. Faradžev,
A. A. Ivanov, M. H. Klin, and A. J. Woldar). Dordrecht,
Netherlands: Kluwer, pp. 459-472, 1994.Ivanov, A. A. "Computation
of Lengths of Orbits of a Subgroup in a Transitive Permutation Group." In Methods
for Complex System Studies. Moscow, Russia: 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.
. It has graph genus 15
(Conder and Stokes 2019).
for 
and 23, respectively (Iofinova and Ivanov 2002).
.
." In Methods for Complex System Studies.
Moscow, Russia: VNIISI, pp. 47-52, 1983.Conder, M.; Malnič,
A.; Marušič, D.; Pisanski, T.; and Potočnik, P. "The Edge-Transitive
but Not Vertex-Transitive Cubic Graph on 112 Vertices." J. Graph Th. 50,
25-42, 2005.