Weighted Cages

TL;DR

This paper introduces the concept of weighted cages, a new variant of regular graphs, providing existence criteria, order determinations for specific girths, Moore-like bounds, and computational insights.

Contribution

It defines weighted cages, characterizes their existence, and offers new bounds and computational results for these graph variants.

Findings

Existence criteria for weighted cages

Order determination for girths 3 and 4

Moore-like bounds established

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.