# 110-vertex Iofinova-Ivanov graph

110-vertex Iofinova-Ivanov graph
Vertices 110
Edges 165
Diameter 7
Girth 10
Automorphisms 1320 (PGL2(11))
Chromatic number 2
Chromatic index 3
Properties semi-symmetric
bipartite
cubic
Hamiltonian
Table of graphs and parameters

The 110-vertex Iofinova-Ivanov graph is, in graph theory, a semi-symmetric cubic graph with 110 vertices and 165 edges.

## Properties

Iofinova and Ivanov proved in 1985 the existence of five and only five semi-symmetric cubic bipartite graphs whose automorphism groups act primitively on each partition.[1] The smallest has 110 vertices. The others have 126, 182, 506 and 990.[2] The 126-vertex Iofinova-Ivanov graph is also known as the Tutte 12-cage.

The diameter of the 110-vertex Iofinova-Ivanov graph, the greatest distance between any pair of vertices, is 7. Its radius is likewise 7. Its girth is 10.

It is 3-connected and 3-edge-connected: to make it disconnected at least three edges, or at least three vertices, must be removed.

### Coloring

The chromatic number of the 110-vertex Iofina-Ivanov graph is 2: its vertices can be 2-colored so that no two vertices of the same color are joined by an edge. Its chromatic index is 3: its edges can be 3-colored so that no two edges of the same color met at a vertex.

### Algebraic properties

The characteristic polynomial of the 110-vertex Iofina-Ivanov graph is ${\displaystyle (x-3)x^{20}(x+3)(x^{4}-8x^{2}+11)^{12}(x^{4}-6x^{2}+6)^{10}}$. The symmetry group of the 110-vertex Iofina-Ivanov is the projective linear group PGL2(11), with 1320 elements.[3]

## Semi-symmetry

Few graphs show semi-symmetry: most edge-transitive graphs are also vertex-transitive. The smallest semi-symmetric graph is the Folkman graph, with 20 vertices, which is 4-regular. The three smallest cubic semi-symmetric graphs are the Gray graph, with 54 vertices, this the smallest of the Iofina-Ivanov graphs with 110, and the Ljubljana graph with 112.[4][5] It is only for the five Iofina-Ivanov graphs that the symmetry group acts primitively on each partition of the vertices.

## References

1. ^ Han and Lu. "Affine primitive groups and Semisymmetric graphs". combinatorics.org. Retrieved 12 August 2015.
2. ^ Weisstein, Eric. "Iofinova-Ivanov Graphs". Wolfram MathWorld. Wolfram. Retrieved 11 August 2015.
3. ^ Iofinova and Ivanov (2013). Investigations in Algebraic Theory of Combinatorial Objects. Springer. p. 470. Retrieved 12 August 2015.
4. ^ Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; Potočnik, P. (2002), "The Ljubljana Graph" (PDF), IMFM Preprints, Ljubljana: Institute of Mathematics, Physics and Mechanics, 40 (845)
5. ^ Conder, Marston; Malnič, Aleksander; Marušič, Dragan; Potočnik, Primož (2006), "A census of semisymmetric cubic graphs on up to 768 vertices", Journal of Algebraic Combinatorics, 23 (3): 255–294, doi:10.1007/s10801-006-7397-3.

## Bibliography

• 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.