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On the diameter and zero forcing number of some graph classes in the Johnson, Grassmann and Hamming association scheme

Aida Abiad, Robin Simoens, Sjanne Zeijlemaker

Abstract

We determine the diameter of generalized Grassmann graphs and the zero forcing number of some generalized Johnson graphs, generalized Grassmann graphs and the Hamming graphs. Our work extends several previously known results.

On the diameter and zero forcing number of some graph classes in the Johnson, Grassmann and Hamming association scheme

Abstract

We determine the diameter of generalized Grassmann graphs and the zero forcing number of some generalized Johnson graphs, generalized Grassmann graphs and the Hamming graphs. Our work extends several previously known results.
Paper Structure (7 sections, 17 theorems, 25 equations, 1 figure)

This paper contains 7 sections, 17 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Diameter and girth of generalized Grassmann graphs
  4. Zero forcing number
  5. Generalized Johnson graphs
  6. Generalized Grassmann graphs
  7. Hamming graphs

Key Result

Lemma 1

For any graph $G$ and field $\mathbb{F}$, $M^{\mathbb{F}}(G) \le Z(G)$.

Figures (1)

  • Figure 2: The zero forcing process for $H(3,4)$. The Hamming graph is depicted as a cube, with each unit subcube representing a vertex. Vertices are adjacent whenever their cubes line up along one of the three main axes.

Theorems & Definitions (27)

  • Lemma 1: AIM, Proposition 2.4 and A, Theorem 2.1
  • Lemma 2: BRESAR2017
  • Lemma 3: bose1966characterization
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • proof
  • Proposition 6
  • proof
  • ...and 17 more