  A bisection of a graph is a bipartition of its vertex set in which the number
of vertices in the two parts differ by at most 1, and its size is the number of
edges which go across the two parts. In this paper, motivated by several
questions and conjectures of Bollob\'as and Scott, we study maximum bisections
of graphs. First, we extend the classical Edwards bound on maximum cuts to
bisections. A simple corollary of our result implies that every graph on $n$
vertices and $m$ edges with no isolated vertices, and maximum degree at most
$n/3 + 1$, admits a bisection of size at least $m/2 + n/6$. Then using the
tools that we developed to extend Edwards's bound, we prove a judicious
bisection result which states that graphs with large minimum degree have a
bisection in which both parts span relatively few edges. A special case of this
general theorem answers a conjecture of Bollob\'as and Scott, and shows that
every graph on $n$ vertices and $m$ edges of minimum degree at least 2 admits a
bisection in which the number of edges in each part is at most $(1/3+o(1))m$.
We also present several other results on bisections of graphs.
