Analogues of Bermond-Bollob\'as Conjecture for Cages Yield Expander Families
<|reference_start|>Analogues of Bermond-Bollob\'as Conjecture for Cages Yield Expander Families: 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.<|reference_end|>
@article{eze2024analogues,
    title={Analogues of Bermond-Bollob\'as Conjecture for Cages Yield Expander
  Families},
    author={Leonard Chidiebere Eze and Robert Jajcay},
    journal={arXiv preprint arXiv:2409.06629},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.06629},
    primaryClass={math.CO cs.DM}
}