[Paper Review] Construction of directed strongly regular graphs using finite incidence structures

Oktay Ölmez, Sung‐Yell Song|arXiv (Cornell University)|Jun 28, 2010

Finite Group Theory Research12 references5 citations

TL;DR

This paper constructs new infinite families of directed strongly regular graphs using finite incidence structures, specifically non-incident point-block pairs in group divisible designs and anti-flags in partial geometries. The key contribution is the realization of previously feasible but unconstructed parameter sets such as (36,12,5,2,5), (54,18,7,4,7), (72,24,10,4,10), (96,24,7,3,7), (108,36,14,8,14), and (108,36,15,6,15), demonstrating their existence via combinatorial constructions.

ABSTRACT

We use finite incident structures to construct new infinite families of directed strongly regular graphs with parameters \[(l(q-1)q^l,\ l(q-1)q^{l-1},\ (lq-l+1)q^{l-2},\ (l-1)(q-1)q^{l-2},\ (lq-l+1)q^{l-2})\] for integers $q$ and $l$ ($q, l\ge 2$), and \[(lq^2(q-1),\ lq(q-1),\ lq-l+1,\ (l-1)(q-1),\ lq-l+1)\] for all prime powers $q$ and $l\in \{1, 2,..., q\}$. The new graphs given by these constructions have parameters $(36, 12, 5, 2, 5)$, $(54, 18, 7, 4, 7)$, $(72, 24, 10, 4, 10)$, $(96, 24, 7, 3, 7)$, $(108, 36, 14, 8, 14)$ and $(108, 36, 15, 6, 15)$ listed as feasible parameters on "Parameters of directed strongly regular graphs," at ${http://homepages.cwi.nl/^\sim aeb/math/dsrg/dsrg.html}$ by S. Hobart and A. E. Brouwer. We review these constructions and show how our methods may be used to construct other infinite families of directed strongly regular graphs.

Motivation & Objective

• To construct new infinite families of directed strongly regular graphs using finite incidence structures such as non-incident point-block pairs and anti-flags.
• To demonstrate the feasibility of previously listed but unconstructed parameter sets for directed strongly regular graphs.
• To extend existing constructions by leveraging divisible designs and partial geometries, particularly anti-flags, to generate new graph families.
• To provide a framework for future exploration of incidence structures in the construction of directed strongly regular graphs.

Proposed method

• Constructs directed strongly regular graphs on the set of non-incident point-block pairs of a group divisible design GD(l, q^{l-2}, q; ql) for integers q ≥ 2 and l ≥ 2.
• Defines edges between pairs (x,S) and (x',S') based on whether x ∈ S', forming a directed graph with specific parameters.
• Uses anti-flags of a partial geometry derived from affine planes to construct a second family of directed strongly regular graphs with parameters (lq²(q−1), lq(q−1), lq−l+1, (l−1)(q−1), lq−l+1).
• Applies a partition-based incidence structure with l disjoint q-element subsets to generate two additional families, one with parameters (ql(l−1), q(l−1), q, 0, q), and another with (ql(l−1), 2q(l−1)−1, ql−1, ql−2, 2q).
• Employs adjacency matrix conditions: A² = tI + λA + μ(J−I−A), and JA = AJ = kJ, to verify the directed strongly regular graph properties.
• Uses Proposition 1.1 to scale known graphs via tensor product with J_m, extending parameter sets to larger graphs.

Experimental results

Research questions

• RQ1 Can new infinite families of directed strongly regular graphs be constructed using non-incident point-block pairs in group divisible designs?
• RQ2 Do anti-flags of partial geometries, particularly those derived from affine planes, yield new directed strongly regular graphs with feasible but previously unconstructed parameters?
• RQ3 Can degenerate incidence structures, such as partitioned sets of points and blocks, generate new families of directed strongly regular graphs?
• RQ4 What is the role of incidence structures like divisible designs and partial geometries in constructing directed strongly regular graphs beyond flags?
• RQ5 How can existing constructions be extended or generalized using combinatorial incidence systems to realize new parameter sets?

Key findings

• The construction using non-incident point-block pairs of GD(l, q^{l-2}, q; ql) yields directed strongly regular graphs with parameters (l(q−1)q^l, l(q−1)q^{l−1}, (lq−l+1)q^{l−2}, (l−1)(q−1)q^{l−2}, (lq−l+1)q^{l−2}) for integers q ≥ 2 and l ≥ 2.
• This construction realizes the feasibility of the parameter set (36,12,5,2,5) when q=3 and l=2, and (96,24,7,3,7) when q=2 and l=4.
• Using anti-flags of a partial geometry from an affine plane of order q, the paper constructs graphs with parameters (lq²(q−1), lq(q−1), lq−l+1, (l−1)(q−1), lq−l+1), realizing (54,18,7,4,7) for q=3, l=2 and (108,36,14,8,14) via Proposition 1.1.
• A third construction based on partitioned sets produces graphs with parameters (ql(l−1), q(l−1), q, 0, q), which are known to be unique for all l ≥ 2 and q ≥ 2.
• A fourth construction yields graphs with parameters (ql(l−1), 2q(l−1)−1, ql−1, ql−2, 2q), matching a family previously constructed by Godsil, Hobart, and Martin, but via a novel incidence-based method.
• The paper confirms the existence of six specific feasible parameter sets listed in Hobart and Brouwer’s database, including (108,36,15,6,15), by constructing explicit graphs for them.