On the analysis of SRG(85, 14, 3, 2)

Zhao, Tianxiao (2024). On the analysis of SRG(85, 14, 3, 2). University of Birmingham. Ph.D.

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Abstract

We investigate the second smallest unresolved feasible set of parameters of strongly regular graphs, (v,k,λ,µ) = (85,14,3,2). We start with the general properties of the 34dimensional Euclidean representation of such a graph G, and then focus on the special subgraphs called segments, which are edge complements in the local subgraphs of G. Since local subgraphs are cubic graphs of order 14, we utilize the available classification of all such cubic graphs. After enumerating possible segments, pairs of segments, and triples of segments, we concentrate on a configuration of four segments build around a maximal 3-clique of G. We enumerate all such configuration and eliminate them case by case, using both combinatorial and metric properties of G. Thus we show that there exists no strongly regular graph with parameters (85,14,3,2). The method we use is quite general and we hope that it can be applied in the future in other unresolved cases with small λ and µ.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):