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New Constructions of Distance-Biregular Graphs

Blas Fernández, Ferdinand Ihringer, Sabrina Lato, Akihiro Munemasa

TL;DR

The paper advances the theory of distance-biregular graphs by (i) introducing a new infinite family tied to a $2$-$Y$-homogeneous structure and a sporadic example linked to Mathon's perp system, (ii) generalizing Delorme's construction to build additional examples via a dual formulation and a derived-hyperoval framework, and (iii) establishing feasibility criteria, parameter-bounds, and a derived-hyperoval methodology that yields new graphs. The authors connect finite geometry, design theory, and algebraic graph theory to expand the catalog of non-regular distance-biregular graphs, including a spectral-excess verification approach and bounds on ambient dimension $n$. They also provide concrete constructions of derived hyperovals and explicit parameter tables for diameter-four cases, along with a new non-existence criterion that narrows the search space. Collectively, the results deepen understanding of how combinatorial designs, finite geometries, and incidence structures give rise to distance-biregular graphs with rich algebraic structure and broad potential applications in related design theories.

Abstract

We construct a new family of distance-biregular graphs related to hyperovals and a new sporadic example of a distance-biregular graph related to Mathon's perp system. The infinite family can be explained using 2-Y- homogeneity, while the sporadic example belongs to a generalization of a construction by Delorme. Additionally, we give a new non-existence criteria for distance-biregular graphs.

New Constructions of Distance-Biregular Graphs

TL;DR

The paper advances the theory of distance-biregular graphs by (i) introducing a new infinite family tied to a --homogeneous structure and a sporadic example linked to Mathon's perp system, (ii) generalizing Delorme's construction to build additional examples via a dual formulation and a derived-hyperoval framework, and (iii) establishing feasibility criteria, parameter-bounds, and a derived-hyperoval methodology that yields new graphs. The authors connect finite geometry, design theory, and algebraic graph theory to expand the catalog of non-regular distance-biregular graphs, including a spectral-excess verification approach and bounds on ambient dimension . They also provide concrete constructions of derived hyperovals and explicit parameter tables for diameter-four cases, along with a new non-existence criterion that narrows the search space. Collectively, the results deepen understanding of how combinatorial designs, finite geometries, and incidence structures give rise to distance-biregular graphs with rich algebraic structure and broad potential applications in related design theories.

Abstract

We construct a new family of distance-biregular graphs related to hyperovals and a new sporadic example of a distance-biregular graph related to Mathon's perp system. The infinite family can be explained using 2-Y- homogeneity, while the sporadic example belongs to a generalization of a construction by Delorme. Additionally, we give a new non-existence criteria for distance-biregular graphs.
Paper Structure (23 sections, 21 theorems, 87 equations, 1 figure)

This paper contains 23 sections, 21 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Distance-Biregular Graphs
  3. Definitions
  4. Generalized Polygons
  5. Notation and Basic Properties
  6. Examples
  7. complete
  8. quasi
  9. srd
  10. largeDiameter
  11. Properties
  12. Parameter Relations and Feasibility
  13. The 2-Y-homogeneous property
  14. Spectral Excess Theorem
  15. Generalization of Delorme's Construction
  16. ...and 8 more sections

Key Result

Theorem 2.1.1

Let $G$ be a graph where every vertex is locally distance-regular. Then $G$ is either distance-regular or distance-biregular.

Figures (1)

  • Figure 1: The subgraph induced by $N_3(z')\cup N_4(z')$.

Theorems & Definitions (32)

  • Theorem 2.1.1: Godsil and Shawe-Taylor distanceRegularised
  • Example 2.2.1
  • Theorem 2.2.2: Mohar and Shawe-Taylor biregularCage
  • Lemma 2.3.1: Delorme delorme
  • Lemma 2.3.2: Delorme delorme
  • Theorem 3.2.1: Delorme delorme, Shawe-Taylor shaweTaylor
  • Example 3.3.1
  • Example 3.3.2
  • Example 3.3.3
  • Example 3.4.1
  • ...and 22 more