scholarly journals Sampling Colorings and Independent Sets of Random Regular Bipartite Graphs in the Non-Uniqueness Region

2022 ◽  
pp. 2198-2207
Author(s):  
Zongchen Chen ◽  
Andreas Galanis ◽  
Daniel Štefankovič ◽  
Eric Vigoda
Keyword(s):  
Independent Sets ◽  
Bipartite Graphs

scholarly journals Neighborhood conditions for balanced independent sets in bipartite graphs

1998 ◽  
Vol 181 (1-3) ◽  
pp. 31-36
Author(s):  
Denise Amar ◽  
Stephan Brandt ◽  
Daniel Brito ◽  
Oscar Ordaz
Keyword(s):  
Independent Sets ◽  
Bipartite Graphs ◽  
Neighborhood Conditions

scholarly journals A Graph Polynomial for Independent Sets of Bipartite Graphs

2012 ◽  
Vol 21 (5) ◽  
pp. 695-714 ◽  
Author(s):  
Q. GE ◽  
D. ŠTEFANKOVIČ
Keyword(s):  
Riemann Hypothesis ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Rational Points ◽  
Perfect Matchings ◽  
Generalized Riemann Hypothesis ◽  
Graph Polynomial ◽  
Exact Evaluation

We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings, the number of perfect matchings, and, for bipartite graphs, the number of independent sets (#BIS).We analyse the complexity of exact evaluation of the polynomial at rational points and show a dichotomy result: for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial.

scholarly journals The Number of Independent Sets in a Regular Graph

2009 ◽  
Vol 19 (2) ◽  
pp. 315-320 ◽  
Author(s):  
YUFEI ZHAO
Keyword(s):  
Regular Graph ◽  
Upper Bound ◽  
Disjoint Union ◽  
Short Proof ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Independence Polynomial ◽  
Bipartite Case

We show that the number of independent sets in an N-vertex, d-regular graph is at most (2d+1 − 1)N/2d, where the bound is sharp for a disjoint union of complete d-regular bipartite graphs. This settles a conjecture of Alon in 1991 and Kahn in 2001. Kahn proved the bound when the graph is assumed to be bipartite. We give a short proof that reduces the general case to the bipartite case. Our method also works for a weighted generalization, i.e., an upper bound for the independence polynomial of a regular graph.

scholarly journals Maximal independent sets in bipartite graphs with at least one cycle

2013 ◽  
Vol Vol. 15 no. 2 (Graph Theory) ◽  
Author(s):  
Shuchao Li ◽  
Huihui Zhang ◽  
Xiaoyan Zhang
Keyword(s):  
Graph Theory ◽  
Independent Set ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Proper Subset ◽  
Maximal Independent Set ◽  
Extremal Graphs ◽  
International Audience ◽  
Maximal Independent Sets ◽  
Disconnected Graphs

Graph Theory International audience A maximal independent set is an independent set that is not a proper subset of any other independent set. Liu [J.Q. Liu, Maximal independent sets of bipartite graphs, J. Graph Theory, 17 (4) (1993) 495-507] determined the largest number of maximal independent sets among all n-vertex bipartite graphs. The corresponding extremal graphs are forests. It is natural and interesting for us to consider this problem on bipartite graphs with cycles. Let \mathscrBₙ (resp. \mathscrBₙ') be the set of all n-vertex bipartite graphs with at least one cycle for even (resp. odd) n. In this paper, the largest number of maximal independent sets of graphs in \mathscrBₙ (resp. \mathscrBₙ') is considered. Among \mathscrBₙ the disconnected graphs with the first-, second-, \ldots, \fracn-22-th largest number of maximal independent sets are characterized, while the connected graphs in \mathscrBₙ having the largest, the second largest number of maximal independent sets are determined. Among \mathscrBₙ' graphs have the largest number of maximal independent sets are identified.

An Entropy Approach to the Hard-Core Model on Bipartite Graphs

2001 ◽  
Vol 10 (3) ◽  
pp. 219-237 ◽  
Author(s):  
JEFF KAHN
Keyword(s):  
Phase Transition ◽  
Bipartite Graph ◽  
Upper Bound ◽  
Hard Core ◽  
Independent Set ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Core Model ◽  
Hard Core Model ◽  
Regular Bipartite Graph

We use entropy ideas to study hard-core distributions on the independent sets of a finite, regular bipartite graph, specifically distributions according to which each independent set I is chosen with probability proportional to λ[mid ]I[mid ] for some fixed λ > 0. Among the results obtained are rather precise bounds on occupation probabilities; a ‘phase transition’ statement for Hamming cubes; and an exact upper bound on the number of independent sets in an n-regular bipartite graph on a given number of vertices.

scholarly journals A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 270
Author(s):  
Igal Sason
Keyword(s):  
Upper Bound ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Theoretic Approach ◽  
Regular Graphs ◽  
Proof Technique ◽  
Information Theoretic ◽  
Theoretic Proof ◽  
Information Theoretic Approach ◽  
General Graphs

This paper studies the problem of upper bounding the number of independent sets in a graph, expressed in terms of its degree distribution. For bipartite regular graphs, Kahn (2001) established a tight upper bound using an information-theoretic approach, and he also conjectured an upper bound for general graphs. His conjectured bound was recently proved by Sah et al. (2019), using different techniques not involving information theory. The main contribution of this work is the extension of Kahn’s information-theoretic proof technique to handle irregular bipartite graphs. In particular, when the bipartite graph is regular on one side, but may be irregular on the other, the extended entropy-based proof technique yields the same bound as was conjectured by Kahn (2001) and proved by Sah et al. (2019).

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