On Computing the Multiplicity of Cycles in Bipartite Graphs Using the Degree Distribution and the Spectrum of the Graph

06/04/2018
by   Ali Dehghan, et al.
0

Counting short cycles in bipartite graphs is a fundamental problem of interest in the analysis and design of low-density parity-check (LDPC) codes. The vast majority of research in this area is focused on algorithmic techniques. Most recently, Blake and Lin proposed a computational technique to count the number of cycles of length g in a bi-regular bipartite graph, where g is the girth of the graph. The information required for the computation is the node degree and the multiplicity of the nodes on both sides of the partition, as well as the eigenvalues of the adjacency matrix of the graph (graph spectrum). In this paper, we extend this result in a number of directions. First, we derive a similar result to compute the number of cycles of length g+2, ..., 2g-2, for bi-regular bipartite graphs. Second, using counter-examples, we demonstrate that the information of the degree distribution and the spectrum of a bi-regular bipartite graph is, in general, insufficient to count the cycles of length i ≥ 2g. Third, we consider irregular bipartite graphs, and show that, in general, the information of degree distribution and spectrum alone is not enough to determine the number of cycles of length i (i-cycles), for any i > g, regardless of the value of g. We demonstrate that the same negative result also holds true for half-regular bipartite graphs, and is also applicable to counting g-cycles for g ≥ 6 and g ≥ 8, in irregular and half-regular graphs, respectively. As positive results, we compute the number of 4-cycles and 6-cycles in irregular and half-regular bipartite graphs, with g ≥ 4 and g ≥ 6, respectively, using only the degree distribution and the spectrum of the graph.

READ FULL TEXT
research
06/04/2018

On Computing the Multiplicity of Short Cycles in Bipartite Graphs Using the Degree Distribution and the Spectrum of the Graph

Counting short cycles in bipartite graphs is a fundamental problem of in...
research
05/31/2019

Cospectral Bipartite Graphs with the Same Degree Sequences but with Different Number of Large Cycles

Finding the multiplicity of cycles in bipartite graphs is a fundamental ...
research
03/20/2019

On Computing the Number of Short Cycles in Bipartite Graphs Using the Spectrum of the Directed Edge Matrix

Counting short cycles in bipartite graphs is a fundamental problem of in...
research
02/08/2021

Complementary cycles of any length in regular bipartite tournaments

Let D be a k-regular bipartite tournament on n vertices. We show that, f...
research
02/15/2019

Effective distribution of codewords for Low Density Parity Check Cycle codes in the presence of disorder

We review the zeta-function representation of codewords allowed by a par...
research
03/10/2023

Simple and efficient four-cycle counting on sparse graphs

We consider the problem of counting 4-cycles (C_4) in a general undirect...
research
07/06/2021

Irregular Invertible Bloom Look-Up Tables

We consider invertible Bloom lookup tables (IBLTs) which are probabilist...

Please sign up or login with your details

Forgot password? Click here to reset