Radio gracefulness of Moore graphs and beyond

The study of radio graceful labelings is motivated by modeling efficient frequency assignment to radio towers, cellular towers, and satellite networks. For a simple, connected graph $G = (V(G), E(G))$, a radio labeling is a mapping f: V(G) → Z⁺ satisfying (for any distinct vertices $u,v$) |f(u)-f(v)| + d(u,v) ≥ diam(G)+1, where $d(u,v)$ is the distance between u and v in G and $diam(G)$ is the diameter of G. A graph is radio graceful if there is a radio labeling such that f(V(G)) = \1, , |V(G)|\. In this paper, we determine the radio gracefulness of low-diameter graphs with connections to high-performance computing, including Moore graphs, bipartite Moore graphs, and approximate Moore graphs like $(r,g)-$cages, Erdős-Rényi polarity graphs, and McKay-Miller-Širáň graphs. We prove a new necessary and sufficient condition for radio graceful bipartite graphs with diameter $3$. We compute the radio number of $(r,g)-$cages arising from generalized $n-$gons. Additionally, we determine Erdős-Rényi polarity graphs and McKay-Miller-Širáň graphs are radio graceful.