The Subgraphs of Order Six of the Family of Strongly Regular Graphs with Parameters $λ=1$ and $μ=2$

TL;DR

The paper addresses the problem of enumerating subgraphs within strongly regular graphs of type $srg(n,k,1,2)$ by deriving a complete account of six-vertex subgraphs. It develops a combinatorial framework that yields 62 distinct six-vertex configurations with counts $n_i$ expressed in terms of $n$, $k$, and a single free parameter $n_3$, and also provides exact formulas for all four- and five-vertex subgraphs. The analysis shows that, while smaller-order subgraphs depend only on $n$ and $k$, six-vertex subgraphs necessitate the extra parameter $n_3$, a feature tied to deep structural constraints and the Conway graph problem $srg(99,14,1,2)$. The work further outlines a path to higher-order subgraphs (order seven) and hints at potential connections to Paley-9 block structures, illustrating how subgraph enumeration informs global graph existence questions in this parameter family.

Abstract

Strongly regular graphs are highly symmetrical and can be described fully with just a few parameters yet the existence of many of them is still under the question. Due to this uncertainty, it is of immense interest to study their structure, in particular to obtain all the possible subgraphs of lower order. In this paper we study the family of strongly regular graphs with parameters $λ=1$ and $μ=2$ and establish all their subgraphs of order six.

Figures (3)

• Figure 1: All possible subgraphs of order six in $srg(n,k,1,2)$.
• Figure 2: All possible subgraphs of order four in $srg(n,k,1,2)$.
• Figure 3: All possible subgraphs of order five in $srg(n,k,1,2)$.

Theorems & Definitions (4)

• Proposition 1
• proof
• Proposition 2
• proof