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On strongly walk regular graphs, triple sum sets and their codes

Michael Kiermaier, Sascha Kurz, Patrick Solé, Michael Stoll, Alfred Wassermann

TL;DR

The work advances the theory of strongly walk-regular graphs (SWRGs) by linking them to coset graphs of duals of projective three-weight codes and by classifying feasible binary and ternary parameters under the key weight-sum constraint. It develops a rigorous Diophantine/eigenvalue framework, showing that for odd $s$, the eigenvalue triples are severely constrained, with explicit results proving only obvious solutions for $s=5$ and $s=7$ and conjecturing the same for all odd $s\ge 9$. The authors combine MacWilliams identities, divisibility techniques, and computer-assisted enumerations to map admissible parameter spaces, and they propose broad conjectures (e.g., $w_2=n/2$ in the binary case under a sum constraint, and a general $q$-ary analogue) supported by small-$a$ bounds and concrete examples. The paper also integrates algebraic geometry (plane curves, elliptic and hyperelliptic curves) and Chabauty methods to derive structural constraints, underscoring the interdisciplinary approach to classifying SWRGs and their associated triple sum sets.

Abstract

Strongly walk regular graphs (SWRGs or $s$-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length~2 are replaced by paths of length~$s$. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an $s$-SWRG has at most 4 distinct eigenvalues $k > θ_1 > θ_2 > θ_3$, and that the triple $(θ_1, θ_2, θ_3)$ satisfies a certain homogeneous polynomial equation of degree $s - 2$ (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for $s = 5$ and $s = 7$, there are only the obvious solutions, and we conjecture this to remain true for all (odd) $s \ge 9$.

On strongly walk regular graphs, triple sum sets and their codes

TL;DR

The work advances the theory of strongly walk-regular graphs (SWRGs) by linking them to coset graphs of duals of projective three-weight codes and by classifying feasible binary and ternary parameters under the key weight-sum constraint. It develops a rigorous Diophantine/eigenvalue framework, showing that for odd , the eigenvalue triples are severely constrained, with explicit results proving only obvious solutions for and and conjecturing the same for all odd . The authors combine MacWilliams identities, divisibility techniques, and computer-assisted enumerations to map admissible parameter spaces, and they propose broad conjectures (e.g., in the binary case under a sum constraint, and a general -ary analogue) supported by small- bounds and concrete examples. The paper also integrates algebraic geometry (plane curves, elliptic and hyperelliptic curves) and Chabauty methods to derive structural constraints, underscoring the interdisciplinary approach to classifying SWRGs and their associated triple sum sets.

Abstract

Strongly walk regular graphs (SWRGs or -SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length~2 are replaced by paths of length~. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an -SWRG has at most 4 distinct eigenvalues , and that the triple satisfies a certain homogeneous polynomial equation of degree (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for and , there are only the obvious solutions, and we conjecture this to remain true for all (odd) .

Paper Structure

This paper contains 12 sections, 24 theorems, 47 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Restrictions on the weights
  4. Weight enumerators and the MacWilliams identity
  5. The coset graph triple sum sets
  6. The geometric point of view to linear codes
  7. Feasible parameters of projective three-weight codes satisfying $\mathbf{w_1+w_2+w_3=3n(q-1)/q}$
  8. Feasible parameters for projective binary three-weight codes with $\mathbf{w_1+w_2+w_3=3n/2}$
  9. Feasible parameters for projective ternary three-weight codes with $\mathbf{w_1+w_2+w_3=2n}$
  10. Plane curves given by the sum of all monomials of given degree
  11. Divisibility for binary linear codes with few weights
  12. Generator matrix of projective three-weights codes satisfying $\mathbf{w_1+w_2+w_3=3(q-1)n/q}$

Key Result

Lemma 1.1

Let $\Gamma$ be a $k$-regular graph with four distinct eigenvalues $k>\theta_1>\theta_2>\theta_3$. Then $\Gamma$ is an $s$-SWRG for $s\ge 3$ if and only if

Theorems & Definitions (43)

  • Lemma 1.1: van Dam, Omidi van2013strongly
  • Lemma 2.1: Bonisoli Bonisoli1984
  • Lemma 2.2: Delsarte delsarte1972weights
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Theorem 2.7
  • proof
  • ...and 33 more