No abstract

Let [Formula: see text] be a finite group and [Formula: see text] be a subgroup of [Formula: see text]. The non-normal graph of [Formula: see text] in [Formula: see text], denoted by [Formula: see text], is defined as the bipartite graph with two parts [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are the normalizer and the core of [Formula: see text] in [Formula: see text], respectively. Two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. In this paper, we consider vertex and edge connectivity of [Formula: see text]. We show that [Formula: see text] and if [Formula: see text] is a positive integer such that [Formula: see text] and [Formula: see text], then the graph [Formula: see text] has a cycle of length [Formula: see text].

Recently, working on the Tanner graph which represents a low density parity check (LDPC) code becomes an interesting research subject. Finding the number of short cycles of Tanner graphs motivated Blake and Lin to investigate the multiplicity of cycles of length girth in bi-regular bipartite graphs, by using the spectrum and degree distribution of the graph. Although there were many algorithms to find the number of cycles, they preferred to investigate in a computational way. Dehghan and Banihashemi counted the number of cycles of length g + 2 and g + 4, where G is a bi-regular bipartite graph and g is the length of the girth G. But they just proposed a descriptive technique to compute the multiplicity of cycles of length less than 2g for biregular bipartite graphs. In this paper, we find the number of cycles of length less than 2g by using spectrum and degree distribution of bi-regular bipartite graphs such that the formula depends only on the partitions of positive integers and the number of closed cycle-free walks from a variable (resp. check) vertex in B c,d and T c,d (resp. T d,c ), which are known.Key words and phrases. (c, d)-regular graph, bipartitel graph, closed walks, cycle-free walk.