[Paper Review] Families of parameters for SRNT graphs

Norman Biggs|ArXiv.org|Nov 12, 2009

Finite Group Theory Research9 references3 citations

TL;DR

This paper refines feasibility conditions for strongly regular graphs with no triangles (SRNT graphs) by introducing simpler divisibility and parity constraints on parameters $q$ and $c$, leading to the identification of infinite families of feasible parameters. It improves the lower bound for the number of vertices $n$, showing $n \geq 4q^3 + 6q^2$ for most $q$, and provides a new interpretation of the Krein bound via the second subconstituent's eigenvalue multiplicities.

ABSTRACT

The feasibility conditions obtained in a previous report are refined, and used to determine several infinite families of feasible parameters for strongly regular graphs with no triangles. The methods are also used to improve the lower bound for the number of vertices, and to derive yet another interpretation of the Krein bound.

Motivation & Objective

• To refine the feasibility conditions for parameters of strongly regular graphs with no triangles (SRNT graphs) by simplifying prior divisibility and parity constraints.
• To establish infinite families of feasible parameter pairs $(q, c)$ that yield valid SRNT graphs.
• To improve the known lower bound for the number of vertices $n$ in such graphs, especially for large $q$.
• To provide a new interpretation of the Krein bound via the eigenvalue multiplicities of the second subconstituent graph $X_2$.

Proposed method

• Derives equivalent feasibility conditions using the integer-valuedness and parity of eigenvalue multiplicities $m_1$ and $m_2$, leading to the requirement that $\alpha = (q^4 - q^2)/c$ and \beta = q(q+1)(q+2)(q+3)/(c+2q) must be integers of the same parity.
• Applies number-theoretic identities to bound the possible values of $q$ for a fixed $c$, showing finiteness except for $c = 2,4,6$, and identifies $q_{\text{max}}$ in terms of $c$ and a derived parameter $h$.
• Uses the formula $n_q(c) = A c + B + D/c$ with $A = (q+1)(q+2)$, $B = 2q^3 + 3q^2 - q$, $D = q^4 - q^2$ to analyze the vertex count and locate minima.
• Analyzes the second subconstituent $X_2$ of an SRNT graph, computing eigenvalue multiplicities $x, y, z$ using trace identities and relating $z$ to the Krein parameter $K_2$ via $qc(c+2q)z = (q^2 + qc + c)K_2$.
• Employs explicit computation and algebraic manipulation to verify feasibility for specific families such as $c = q$, $c = q(q-1)/2$, and $c = q(q+1)$, confirming infinite families.

Experimental results

Research questions

• RQ1 Which parameter pairs $(q, c)$ yield feasible strongly regular graphs with no triangles, and how can the feasibility conditions be simplified?
• RQ2 Are there infinite families of feasible parameters $(q, c)$ for fixed $c$, and if so, for which $c$ do they exist?
• RQ3 Can the lower bound for the number of vertices $n$ in SRNT graphs be improved, and under what conditions is the bound tight?
• RQ4 How does the eigenvalue structure of the second subconstituent $X_2$ relate to the Krein bound, and what does this imply for graph realizability?

Key findings

• The feasibility of parameters $(q, c)$ for SRNT graphs is equivalent to $\alpha = (q^4 - q^2)/c$ and $\beta = q(q+1)(q+2)(q+3)/(c+2q)$ being integers of the same parity.
• Infinite families of feasible parameters exist for $c = 2, 4, 6$, $c = q$, $c = q(q-1)/2$, $c = q(q-1)$, and $c = q(q+1)$, all valid for all $q$ satisfying $k \geq 3$ and $k > c \geq 1$.
• The lower bound for $n$ is improved to $n \geq 4q^3 + 6q^2$ for all $q$ except $q = 2,4,5,6,7,8,12,47$, where it is $4q^3 + 6q^2 - 2(q+1)$.
• For $q = 4,6,8$, the lower bound is $4q^3 + 6q^2 - 2(q-1) + 12/(q-2)$, attained when $c = q-2$, and this value is only feasible for those $q$ where $q-2$ divides 12.
• The eigenvalue multiplicity $z$ of $-(q+c)$ in the second subconstituent $X_2$ is a non-negative integer if and only if the Krein parameter $K_2 \geq 0$, providing a new interpretation of the Krein bound.
• When $c = q(q+1)$, the multiplicity $z = 0$, implying $X_2$ is itself an SRNT graph, and the lower bound $n = 4q^3 + 6q^2$ is achieved.