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.
|Title of host publication||Connections in Discrete Mathematics|
|Subtitle of host publication||A Celebration of the Work of Ron Graham|
|Publisher||Cambridge University Press|
|Number of pages||21|
|State||Published - 1 Jan 2018|