Equatorial graph existence conjecture

Let an equatorial graph be a graph attaining the lower bound for the order of a $(\delta+,g,q)$-graph, where $\delta$ is its degree and $g$ its girth. A Moore graph is a graph attaining the Moore bound with degree $\delta$ and girth $g$.

Equatorial graph existence conjecture. There exists an equatorial graph with degree $\delta$ and girth $g$ if and only if there exists a Moore graph with degree $\delta$ and girth $g$.

This conjecture asks whether equatorial graphs exist for exactly the same degree and girth parameters as Moore graphs. The paper establishes structural results for equatorial graphs and characterizes several cases, but does not resolve the general existence question.