[Paper Review] Regular Bipartite Graphs And Their Properties

Vivek S. Nittoor|arXiv (Cornell University)|Dec 31, 2012

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TL;DR

This paper introduces a novel symmetric permutation tree (SPT(m)) abstraction to systematically enumerate labeled and non-isomorphic r-regular bipartite graphs, particularly focusing on 2-regular cases. It presents a complete algorithm for generating all non-isomorphic 2-regular bipartite graphs with 2m vertices and derives formulae for compatible permutations and automorphism groups using partition-based enumeration, proving the maximum girth of such graphs is 2m.

ABSTRACT

We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all non-isomorphic 2-regular bipartite graphs for a specified number of vertices has been described and a mathematical proof has been provided for its completeness. An abstraction of m Symmetric Permutation Tree in order to visualize a labeled r-Regular Bipartite Graph with 2m vertices and enumerate its automorphism group has been introduced. An algorithm to generate the partition associated with two compatible permutations has been introduced. The relationship between Automorphism Group and permutation enumeration problem has been used to derive formulae for the number of compatible permutations corresponding to a specified partition.

Motivation & Objective

• To develop a general algorithm for exhaustive enumeration of non-isomorphic 2-regular bipartite graphs with 2m vertices.
• To introduce the symmetric permutation tree (SPT(m)) as a visualization and enumeration tool for labeled r-regular bipartite graphs.
• To establish a mathematical framework linking automorphism groups and permutation enumeration for regular bipartite graphs.
• To derive formulae for the number of compatible permutations corresponding to a given partition of vertex cycles.
• To determine the maximum possible girth of 2-regular bipartite graphs with 2m vertices.

Proposed method

• Introduces a symmetric permutation tree (SPT(m)) to represent and enumerate all permutations of m elements, forming the basis for graph enumeration.
• Uses recursive backtracking via procedure P to exhaustively generate all m! permutations, ensuring completeness via proof of one-to-one correspondence with leaf nodes.
• Applies bi-adjacency matrix representation to model labeled r-regular bipartite graphs, where each row and column sums to r.
• Defines isomorphism between graphs via row and column permutations of the bi-adjacency matrix, enabling non-isomorphic graph filtering.
• Derives formulae for compatible permutations based on cycle partitions β ∈ P₂(m), linking permutation structure to graph automorphism groups.
• Employs the Known Cycle Theorem to relate cycle lengths in 2-regular graphs to partition components, with cycle length 2qᵢ for each component qᵢ.

Experimental results

Research questions

• RQ1 What is the complete set of non-isomorphic 2-regular bipartite graphs with 2m vertices, and how can they be systematically enumerated?
• RQ2 How can the automorphism group of a labeled r-regular bipartite graph be enumerated and characterized using permutation structures?
• RQ3 What is the relationship between compatible permutations and cycle partitions in regular bipartite graphs?
• RQ4 What is the maximum possible girth of a 2-regular bipartite graph with 2m vertices?
• RQ5 Can a general algorithm be constructed to enumerate all non-isomorphic r-regular bipartite graphs for arbitrary r and m?

Key findings

• The algorithm ESPT(m) correctly enumerates all m! permutations of m elements, proven via one-to-one correspondence between leaf nodes and permutations.
• The maximum attainable girth of a 2-regular bipartite graph with 2m vertices is 2m, achieved when the graph is a single cycle of length 2m.
• For a 2-regular bipartite graph isomorphic to Ψ(β), the cycle lengths are exactly {2qᵢ} where qᵢ are components of the partition β ∈ P₂(m).
• The number of compatible permutations corresponding to a given partition β is derived using the relationship between automorphism groups and permutation enumeration.
• The symmetric permutation tree SPT(m) provides a complete and systematic framework for visualizing and enumerating labeled r-regular bipartite graphs and their symmetries.
• The paper establishes a foundational framework for searching r-regular bipartite graphs with maximum girth, particularly for r = 2 and r = 3.