Strongly regular graphs with parameters (85,14,3,2) do not exist

TL;DR

This work resolves the unresolved SRG case $\mathrm{srg}(85,14,3,2)$ by combining a rigorous local-structure analysis with an exhaustive four-step computational enumeration. Local neighbourhoods are restricted to 39 good cubic graphs on 14 vertices, yielding maximal 3-cliques and a decomposition of the neighbourhood around a clique into three segments, $S_x,S_y,S_z$, whose gluing is systematically explored via Gram-matrix constraints in the Euclidean representation with cosine sequence $(1,\tfrac{2}{7},-\tfrac{1}{14})$. Through iterative SPD checks using LDLT, core-matchings, and a recursive addition of eight extra neighbours followed by exact-set verification, every candidate configuration is eliminated, proving non-existence. The result showcases a scalable, high-precision approach for resolving long-standing unresolved SRG parameter sets, leveraging both combinatorial structure and linear-algebraic feasibility criteria, and highlights the potential for applying similar methodology to other near-miss SRG cases.

Abstract

We investigate the second smallest unresolved feasible set of parameters of strongly regular graphs, $(v,k,λ,μ)=(85,14,3,2)$. Using the classification of cubic graphs of small degree, we restrict possible local structure of such a graph $G$. After that, we exhaustively enumerate possible neighbourhoods of a maximal $3$-clique of $G$ and check them against a variety of conditions, including the combinatorial ones, coming from $λ=3$ and $μ=2$, as well as the linear algebra ones, utilising the Euclidean representation of $G$. These conditions yield contradiction in all cases, and hence, no $\mathrm{srg}(85,14,3,2)$ exists.

Figures (2)

• Figure 1: Relevant cubic graphs of order 10
• Figure 2: The composition of $T$

Theorems & Definitions (45)

• Theorem 1.1
• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Corollary 2.4
• Theorem 2.5
• Theorem 2.6: BrouwerDRGbook, Section 4.1B
• Corollary 2.7
• Lemma 3.1
• proof