Symmetric Bipartite Graphs and Graphs with Loops

TL;DR

This paper proves that bipartite graphs with equal degree sequences in both parts can be symmetrically constructed with an involutive automorphism, linking symmetric bipartite graphs to graphs with loops.

Contribution

It establishes a connection between symmetric bipartite graphs and graphs with loops, providing a method to construct symmetric bipartite graphs with given degree sequences.

Findings

Existence of symmetric bipartite graphs with equal degree sequences

Relationship between symmetric bipartite graphs and graphs with loops

Construction method for symmetric bipartite graphs

Abstract

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.