Analogues of Bermond-Bollob\'as Conjecture for Cages Yield Expander Families

TL;DR

This paper explores a theoretical link between cages and expander graphs by adapting Bermond and Bollobás Conjecture variants, suggesting that solutions could lead to new expander graph families.

Contribution

It introduces three variants of the Bermond and Bollobás Conjecture for cages and shows their potential to generate expander graph families, establishing a novel connection.

Findings

Proposes three conjecture variants for cages.

Shows that solving these conjectures could produce expander families.

Establishes a theoretical link between cages and expanders.

Abstract

This paper presents a possible link between Cages and Expander Graphs by introducing three interconnected variants of the Bermond and Bollob\'as Conjecture, originally formulated in 1981 within the context of the Degree/Diameter Problem. We adapt these conjectures to cages, with the most robust variant posed as follows: Does there exist a constant $c$ such that for every pair of parameters $(k,g)$ there exists a $k$-regular graph of girth $g$ and order not exceeding $ M(k,g) + c $?; where $M(k,g)$ denotes the value of the so-called Moore bound for cages. We show that a positive answer to any of the three variants of the Bermond and Bollob\'as Conjecture for cages considered in our paper would yield expander graphs (expander families); thereby establishing a connection between Cages and Expander Graphs.