Links between two semisymmetric graphs on 112 vertices through the lens of association schemes

One of the most striking impacts between geometry, combinatorics and graph theory, on one hand, and algebra and group theory, on the other hand, arise from a concrete necessity to manipulate with the symmetry of the investigated objects. In the case of graphs, we talk about such tasks as identification and compact representation of graphs, recognition of isomorphic graphs and computation of automorphism groups of graphs. Different classes of interesting graphs are defined in terms of the level of their symmetry, which which is founded on the concept of transitivity. A semisymmetric graph, the central subject in this paper, is a regular graph Γ such that its automorphism group Aut(Γ) acts transitively on the edge set E(Γ), though intransitively on the vertex set V (Γ). More specifically, we investigate links between two semisymmetric graphs, both on 112 vertices, the graph N of valency 15 and the graph L of valency 3. While the graph N can be easily defined and investigated without serious computations, the manipulation with L inherently depends on a quite heavy use of a computer. The graph L, commonly called the Ljubljana graph, see [76], has quite a striking history, due to efforts of a few generations of mathematicians starting from M. C. Gray and I. Z. Bouwer and ending with M. Conder, T. Pisanski and their colleagues; the order of its automorphism group is 168. The graph L turns out to be a spanning subgraph of the graph N , which has the group S8 of order 8! as automorphism group. In our paper we reveal numerous interesting links between L and N , as well as with diverse combinatorial structures, including association schemes, coherent algebras, symmetric configurations, overlarge sets of Fano planes, partial geometries, etc. For this purpose we exploit tools from algebraic graph theory