  Let G be a graph obtained by taking r>=2 paths and identifying all first
vertices and identifying all the last vertices. We compute the
Castelnuovo--Mumford regularity of the quotient S/I(X), where S is the
polynomial ring on the edges of G and I(X) is the vanishing ideal of the
projective toric subset parameterized by G. The case we consider is the first
case where the regularity was unknown, following earlier computations (by
several authors) of the regularity when G is a tree, cycle, complete graph or
complete bipartite graph, but specially in light of the reduction of the
computation of the regularity in the bipartite case to the computation of the
regularity of the blocks of G. We also prove new inequalities relating the
Castelnuovo--Mumford regularity of S/I(X) with the combinatorial structure of
G, for a general graph.
