Induced Subgraphs of Order Seven and Their Frequencies in $srg(n,k,1,2)$

Abstract: In this paper, we examine the structure of strongly regular graphs with parameters $λ= 1$ and $μ= 2$. In particular, we provide a complete classification of induced subgraphs of order seven and determine their relative frequencies. These findings contribute to a finer understanding of the local structure of such graphs and may be useful in related combinatorial and algebraic investigations.

• The paper classifies exactly 208 induced subgraphs on seven vertices in every strongly regular graph with parameters srg(n,k,1,2), completing the order-seven census beyond Hamiltonian cases.
• The paper derives explicit polynomial frequency formulas for all configurations using n and k plus two free structural parameters, n₃ and z₁₁, through combinatorial double-counting.
• The paper identifies 15 configuration frequencies determined solely by n and k, providing potential divisibility, integrality, and non-existence tests for problems such as the Conway graph srg(99,14,1,2).

Background and motivation

Strongly regular graphs, defined by the parameter quadruple $(n,k,\lambda,\mu)$ and equivalently by having exactly three eigenvalues with $k$ of multiplicity one, have been a central object in algebraic graph theory since Bose's foundational work. The subfamily with $\lambda = 1$ and $\mu = 2$ is of particular interest because it contains several long-standing open existence problems, most prominently the $srg(99,14,1,2)$ — the 99-vertex Conway graph problem, listed among Conway's $1,000 problems. Known members of the family are sparse: apart from small valencies$k = 2$and$k = 4$, the only other known example has valency$k = 22$, derived from the perfect ternary Golay code by Berlekamp, Van Lint, and Seidel.</p> <p>Prior work on this family has proceeded largely through local structural analysis. Wilbrink and Brouwer&#39;s lemma was used to exclude a $(57,14,1)$$graph; Lou and Murin established a forbidden induced subgraph of order nine for$$k = 14$; Makhnev and Minkova studied automorphism groups; and the author&#39;s earlier papers developed lower bounds on hexagon counts, a complete enumeration of induced subgraphs of order six, and an enumeration of Hamiltonian induced subgraphs of order seven. The present paper completes that program at order seven: it classifies <strong>all</strong> induced subgraphs on seven vertices (not only the Hamiltonian ones) and derives closed-form expressions for their frequencies.</p> <h2 class='paper-heading' id='main-contribution'>Main contribution</h2> <p>The paper establishes two results:</p> <ol> <li><strong>Complete classification</strong>: there are exactly 208 distinct induced subgraphs of order seven in any $k$0, all depicted in the paper's main figure.

Frequency formulas: each subgraph $k$1 has a frequency $k$2 given as an explicit polynomial expression in $k$3 and $k$4, together with corrections involving exactly two free parameters, $k$5 and $k$6.

The free variables reflect genuine indeterminacy: the frequencies cannot be pinned down from the global parameters alone. Here $k$7 follows the convention of the author's order-six enumeration, and $k$8 denotes the frequency of the specific seven-vertex configuration $k$9. Consequently, some configurations may be absent (frequency zero) depending on the values these variables take in a particular graph.

The formulas themselves range from simple linear expressions — e.g., $\lambda = 1$0 and $\lambda = 1$1 — to polynomials of degree up to eight in $\lambda = 1$2 multiplied by $\lambda = 1$3, as in $\lambda = 1$4, which carries a degree-nine factor divided by $\lambda = 1$5. The structure of the formulas is systematic: leading terms count embeddings determined purely by local regularity constraints ($\lambda = 1$6, $\lambda = 1$7 force factors such as $\lambda = 1$8, $\lambda = 1$9, $\mu = 2$0, $\mu = 2$1), while the $\mu = 2$2 and $\mu = 2$3 terms correct for configurations containing dense or special local structures whose counts are not globally forced.

A notable feature is that many frequencies are determined — independent of both free variables — including $\mu = 2$4, $\mu = 2$5, $\mu = 2$6, $\mu = 2$7, $\mu = 2$8, $\mu = 2$9, $srg(99,14,1,2)$0, $srg(99,14,1,2)$1, $srg(99,14,1,2)$2, $srg(99,14,1,2)$3, $srg(99,14,1,2)$4, $srg(99,14,1,2)$5, $srg(99,14,1,2)$6, $srg(99,14,1,2)$7, and $srg(99,14,1,2)$8. These rigid counts are functions of $srg(99,14,1,2)$9 alone and hold uniformly across every graph in the family, making them candidate invariants for non-existence arguments via divisibility or integrality conditions.

Methodological remarks

The derivations follow the counting techniques established in the author's previous work on order-six and Hamiltonian order-seven subgraphs; the paper explicitly omits rederiving them, noting they are analogous to prior treatments. Algebraic simplification of the resulting expressions was performed with Wolfram Alpha. This is a purely combinatorial double-counting approach; no spectral or computational search over actual graphs is involved, so the formulas are valid for any graph realizing the parameters, whether or not such graphs are known to exist for a given $1,000 problems. Known members of the family are sparse: apart from small valencies$0.

Limitations and open questions

Several caveats should be noted plainly. First, the frequency formulas are parametric rather than absolute: without independent determination of $1,000 problems. Known members of the family are sparse: apart from small valencies$1 and $1,000 problems. Known members of the family are sparse: apart from small valencies$2, most of the 208 frequencies remain undetermined, and the paper does not provide bounds or integrality constraints on these free variables. Second, the paper does not itself derive any new non-existence result; its utility for problems such as the $1,000 problems. Known members of the family are sparse: apart from small valencies$3 is asserted prospectively, on the grounds that forbidden-subgraph arguments (as in the Wilbrink–Brouwer and Lou–Murin results) can be sharpened when small-subgraph frequencies are known. Whether the new formulas yield concrete contradictions for specific parameter values — for instance, negative predicted frequencies or failed divisibility conditions at $1,000 problems. Known members of the family are sparse: apart from small valencies$4 — is left unexamined. Third, the classification is presented pictorially in a large figure without accompanying machine-readable data, which may complicate verification and reuse. Finally, extending the approach to order eight appears combinatorially explosive, and no path toward it is indicated.

Conclusion

This paper completes the enumeration of small induced subgraphs in $1,000 problems. Known members of the family are sparse: apart from small valencies$5 through order seven, providing all 208 configurations and explicit frequency formulas governed by two free structural parameters. The principal value of the work lies in supplying a fine-grained local census that future non-existence and construction efforts — particularly for the Conway $1,000 problems. Known members of the family are sparse: apart from small valencies$6 problem — may exploit; converting that census into actual feasibility constraints remains the natural open task.