  A balanced graph is a bipartite graph with no induced circuit of length 2 mod
4. These graphs arise in linear programming. We focus on graph-algebraic
properties of balanced graphs to prove a complete classification of balanced
Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, we prove
that there is no cubic balanced planar graph. Finally, some remarkable
conjectures for balanced regular graphs are also presented.
