  The degree-diameter problem for graphs is to find the largest number of
vertices a graph can have given its diameter and maximum degree. We show we can
realize this problem in terms of quasigroups, 1-factors and permutation groups.
Our investigation originated from the study of graphs as the Cayley graphs of
groupoids with d generators, a left identity and right cancellation; that is, a
right quasigroup. This enables us to provide compact algebraic definitions for
some important graphs that are either given as explicit edge lists or as the
Cayley coset graphs of groups larger than the graph. One such example is a
single expression for the Hoffman-Singleton graph. From there, we notice that
the groupoids can be represented uniquely by a set of disjoint permutations and
we explore the consequences of that observation.
