Maximum algebraic connectivity implies maximum girth

Let $G$ be a regular graph of degree $d$ and order $n$. Its algebraic connectivity is denoted by $AC(G)$, and its girth is the length of its shortest cycle.

Maximum-connectivity–girth conjecture. For fixed degree $d$ and order $n$, a graph maximizing $AC(G)$ also has maximum possible girth.

This conjecture is motivated by computational evidence from complete lists of cubic and quartic graphs for small orders and underlies the search for diameter-maximal graphs. It is stated as an open problem.