The biMoore graph conjecture from perfect difference sets

Let $s$ be a positive integer with $s-1$ a prime power, let $S$ be a perfect difference set of size $s$, and let $G_m(S)$ be the graph constructed in Corollary. A bipartite graph is $(r,s;3)$-biMoore when it has degrees $(r,s)$ and diameter $3$, and attains the corresponding bipartite Moore bound. BiMoore graph conjecture. For $s-1$ a prime power, the graphs $G_m(S)$ of Corollary are $(ms,s;3)$-biMoore graphs. The preceding construction supplies such graphs from perfect difference sets; the conjecture asserts that they attain the Moore bound, while the broader existence and construction theory for suitable difference sets remains open.