  In this paper, we give a lower bound for the maximum and minimum genus of a
multibranched surface by the first Betti number and the minimum and maximum
genus of the boundary of the neighborhood of it, respectively. As its
application, we show that the maximum and minimum genus of $G\times S^1$ is
equal to twice of the maximum and minimum genus of $G$ for a graph $G$,
respectively. This provides an interplay between graph theory and 3-manifold
theory.

Geometric Topology;Combinatorics