We show that if the two parts of a finite bipartite graph have the same
degree sequence, then there is a bipartite graph, with the same degree
sequences, which is symmetric, in that it has an involutive graph automorphism
that interchanges its two parts. To prove this, we study the relationship
between symmetric bipartite graphs and graphs with loops.Comment: arXiv admin note: substantial text overlap with arXiv:1302.365
