Unramified Graph Covers of Finite Degree

Regarding the fundamental group of a finite connected undirected graph X as the absolute Galois group of X, in this chapter we explore graph theoretical counterparts of several important theorems for number fields. We first characterize finite-degree unramified normal covers of X for which the Chebotarëv density theorem holds in natural density. Then we give finite necessary and sufficient conditions to classify finite-degree unramified covers of X up to equivalence. Similar to the reciprocity law for finite Galois extensions of a number field, it is shown that the unramified normal covers of X of degree d, up to isomorphism, are determined by the primes of X of length ≤ (4|X| - 1)d - 1 which split completely. Finally we obtain a finite criterion for Sunada equivalence, improving a result of Somodi.