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Weighted Cages

G. Araujo-Pardo, C. De la Cruz, M. Matamala, M. A. Pizaña

Abstract

Cages ($r$-regular graphs of girth $g$ and minimum order) and their variants have been studied for over seventy years. Here we propose a new variant, "weighted cages". We characterize their existence; for cases $g=3,4$ we determine their order; we give Moore-like bounds and present some computational results.

Weighted Cages

Abstract

Cages (-regular graphs of girth and minimum order) and their variants have been studied for over seventy years. Here we propose a new variant, "weighted cages". We characterize their existence; for cases we determine their order; we give Moore-like bounds and present some computational results.

Paper Structure

This paper contains 9 sections, 19 theorems, 25 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Terminology and Preliminaries
  3. Existence of weighted cages
  4. Moore-like bounds
  5. Weighted cages of girth 3
  6. Weighted cages of girth 4
  7. Weighted cages of girth 5 and 6
  8. Experimental results
  9. General constructions for $(a,b,g)$-wcages

Key Result

Lemma 2.1

Let $a,b\geq 0$ and $g\geq 3$. Then $n(a,b,g)\geq a+b+1$. Moreover, if $ab \equiv 1$, then $n(a,b,g)\geq a+b+2$.

Figures (5)

  • Figure 1: Construction in the proof of Theorem \ref{['construction']}.
  • Figure 2: Moore-like wtree for $a=2, b=2, g=9$.
  • Figure 3: Moore-like wtrees for $a=2, b=2, g=8$.
  • Figure 4: Construction of (a,b,4)-wcages for $n=\frac{5a}{2}+k$.
  • Figure 5: Construction of (a,b,4)-wcages for $n=\frac{5a}{2}+\frac{a}{2}r+k$ with $r=4$.

Theorems & Definitions (36)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 26 more