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arXiv · 2409.06629

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

Leonard Chidiebere Eze·Robert Jajcay
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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.

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Leonard Chidiebere Eze, Robert Jajcay. 2024-09-10. Analogues of Bermond-Bollob\'as Conjecture for Cages Yield Expander Families. https://arxiv.org/abs/2409.06629

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