Automorphism groups of some families of bipartite graphs - K.G. Sreekumar; K. Manilal (Electronic Journal of Graph Theory and Applications (EJGTA), 2021)
This paper discusses the automorphism group of a  class of  weakly semiregular bipartite graphs and its subclass called WSBEND graphs.  It also tries to analyse the  automorphism group of the SM sum graphs and SM balancing graphs.  These graphs  are weakly semiregular bipartite graphs too.  The SM sum graphs  are particular cases  of bipartite Kneser graphs. The  bipartite Kneser type graphs are defined on n-sets for a fixed positive integer n. The  automorphism groups of the bipartite Kneser type graphs are related to that of weakly semiregular bipartite graphs.  Weakly semiregular bipartite  graphs  in which   the neighbourhoods of the vertices in the SD part having  the same degree sequence, possess  non trivial automorphisms.  The automorphism groups of SM sum graphs are isomorphic to the  symmetric groups. The relationship between the  automorphism groups of SM balancing  graphs  and  symmetric  groups are established here.   It has been observed by using the well known algorithm Nauty, that the size of automorphism  groups of SM balancing graphs are prodigious.  Every weakly semiregular bipartite graphs with  k-NSD subparts has a matching which saturates the smaller partition.