  Given an arbitrary graph G, we study its basic covers algebra, which is the
symbolic fiber cone of the Alexander dual of the edge ideal of G. Extending
results of Villarreal and Benedetti-Constantinescu-Varbaro, valid only in the
case when G is bipartite, we characterize in a combinatorial fashion the
situations when: 1) the basic covers algebra is a domain, and 2) it is a domain
and in addition (the edge ideal of) G is unmixed. It turns out that the last
result gives a complete characterization of those graphs for which any symbolic
power of the edge ideal is generated by monomials of the same degree.
