110-vertex Iofinova-Ivanov graph
# 110-vertex Iofinova-Ivanov graph

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. The smallest has 110 vertices. The others have 126, 182, 506 and 990. 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 $(x-3)x^{20}(x+3)(x^{4}-8x^{2}+11)^{12}(x^{4}-6x^{2}+6)^{10}$. The symmetry group of the 110-vertex Iofina-Ivanov is the projective linear group $PGL_2(11),$ with 1320 elements.

## 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. It is only for the five Iofina-Ivanov graphs that the symmetry group acts primitively on each partition of the vertices.

