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On the existence of small strictly Neumaier graphs

Aida Abiad, Maarten De Boeck, Sjanne Zeijlemaker

Abstract

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this work, we prove several results on the existence of small strictly Neumaier graphs. In particular, we present a theoretical proof of the uniqueness of the smallest strictly Neumaier graph with parameters $(16,9,4;2,4)$, we establish the existence of a strictly Neumaier graph with parameters $(25,12,5;2,5)$, and we disprove the existence of strictly Neumaier graphs with parameters $(25,16,9;3,5)$, $(28,18,11;4,7)$, $(33,24,17;6,9)$, $(35,22,12;3,5)$ and $(55,34,18;3,5)$. Our proofs use combinatorial techniques and a novel application of integer programming methods.

On the existence of small strictly Neumaier graphs

Abstract

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this work, we prove several results on the existence of small strictly Neumaier graphs. In particular, we present a theoretical proof of the uniqueness of the smallest strictly Neumaier graph with parameters , we establish the existence of a strictly Neumaier graph with parameters , and we disprove the existence of strictly Neumaier graphs with parameters , , , and . Our proofs use combinatorial techniques and a novel application of integer programming methods.
Paper Structure (7 sections, 10 theorems, 4 equations, 18 figures, 2 tables)

This paper contains 7 sections, 10 theorems, 4 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A proof for the uniqueness of the smallest strictly Neumaier graph
  4. A new strictly Neumaier graph on 25 vertices
  5. A new approach for nonexistence
  6. Integer linear programming and the graph complement
  7. Nonexistence results of small strictly Neumaier graphs

Key Result

Theorem 1

Let $\Gamma$ be an edge-regular graph with parameters $(v,k,\lambda)$. Then

Figures (18)

  • Figure 1: The diamond graph, paw graph and bowtie graph.
  • Figure 2: The graph $K_{1,4}\cup 4K_1$ and its vertex labeling.
  • Figure 3: Construction of all diamond-free 4-regular graphs on 9 vertices if $\Gamma(x)$ is an independent set.
  • Figure 4: Construction of all diamond-free 4-regular graphs on 9 vertices if $\Gamma(x)$ contains one edge.
  • Figure 5: Construction of all diamond-free 4-regular graphs on 9 vertices if $\Gamma(x)$ contains two edges.
  • ...and 13 more figures

Theorems & Definitions (15)

  • Theorem 1: bcn,ABIAD2023105684
  • Theorem 2: N1981, Theorem 1.1
  • Theorem 3: N1981 and evans2018smallest
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 5 more