  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.
