Analogues of Bermond-Bollobás 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 Bollobás 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 val