# Ljubljana graph

Ljubljana graph
The Ljubljana graph as a covering graph of the Heawood graph
Vertices 112
Edges 168
Diameter 8
Girth 10
Automorphisms 168
Chromatic number 2
Chromatic index 3
Properties Cubic
Semi-symmetric
Hamiltonian

In the mathematical field of graph theory, the Ljubljana graph is an undirected bipartite graph with 112 vertices and 168 edges.[1]

It is a cubic graph with diameter 8, radius 7, chromatic number 2 and chromatic index 3. Its girth is 10 and there are exactly 168 cycles of length 10 in it. There are also 168 cycles of length 12.[2]

## Construction

The Ljubljana graph is Hamiltonian and can be constructed from the LCF notation : [47, -23, -31, 39, 25, -21, -31, -41, 25, 15, 29, -41, -19, 15, -49, 33, 39, -35, -21, 17, -33, 49, 41, 31, -15, -29, 41, 31, -15, -25, 21, 31, -51, -25, 23, 9, -17, 51, 35, -29, 21, -51, -39, 33, -9, -51, 51, -47, -33, 19, 51, -21, 29, 21, -31, -39]2.

The Ljubljana graph is the Levi graph of the Ljubljana configuration, a quadrangle-free configuration with 56 lines and 56 points.[2] In this configuration, each line contains exactly 3 points, each point belongs to exactly 3 lines and any two lines intersect in at most one point.

## Algebraic properties

The automorphism group of the Ljubljana graph is a group of order 168. It acts transitively on the edges the graph but not on its vertices : there are symmetries taking every edge to any other edge, but not taking every vertex to any other vertex. Therefore, the Ljubljana graph is a semi-symmetric graph, the third smallest possible cubic semi-symmetric graph after the Gray graph on 54 vertices and the Iofinova-Ivanov graph on 110 vertices.[3]

The characteristic polynomial of the Ljubljana graph is

${\displaystyle (x-3)x^{14}(x+3)(x^{2}-x-4)^{7}(x^{2}-2)^{6}(x^{2}+x-4)^{7}(x^{4}-6x^{2}+4)^{14}.\ }$

## History

The Ljubljana graph was first published in 1993 by Brouwer, Dejter and Thomassen[4] as a self-complementary subgraph of the Dejter graph. [5]

In 1972, Bouwer was already talking of a 112-vertices edge- but not vertex-transitive cubic graph found by R. M. Foster, nonetheless unpublished.[6] Conder, Malnič, Marušič, Pisanski and Potočnik rediscovered this 112-vertices graph in 2002 and named it the Ljubljana graph after the capital of Slovenia.[2] They proved that it was the unique 112-vertices edge- but not vertex-transitive cubic graph and therefore that was the graph found by Foster.

## References

1. ^
2. ^ a b c Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; and Potočnik, P. "The Ljubljana Graph." 2002. [1].
3. ^ Marston Conder, Aleksander Malnič, Dragan Marušič and Primž Potočnik. "A census of semisymmetric cubic graphs on up to 768 vertices." Journal of Algebraic Combinatorics: An International Journal. Volume 23, Issue 3, pages 255-294, 2006.
4. ^ Brouwer, A. E.; Dejter, I. J.; and Thomassen, C. "Highly Symmetric Subgraphs of Hypercubes." J. Algebraic Combinat. 2, 25-29, 1993.
5. ^ Klin M.; Lauri J.; Ziv-Av M. "Links between two semisymmetric graphs on 112 vertices through the lens of association schemes", Jour. Symbolic Comput., 47–10, 2012, 1175–1191.
6. ^ Bouwer, I. A. "On Edge But Not Vertex Transitive Regular Graphs." J. Combin. Th. Ser. B 12, 32-40, 1972.