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Entropy, Graph Homomorphisms, and Dissociation Sets.

Ziyuan Wang1, Jianhua Tu1, Rongling Lang2

  • 1School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China.

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|January 21, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces an entropy approach to establish upper bounds for graph homomorphisms and dissociation sets in bipartite graphs. These findings offer new insights into graph theory and combinatorial structures.

Keywords:
bipartite graphsdissociation setsentropygraph homomorphismsindependent sets

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Area of Science:

  • Graph Theory
  • Combinatorics
  • Theoretical Computer Science

Background:

  • Graph homomorphisms and dissociation sets generalize independent sets.
  • Understanding these structures is crucial in various areas of mathematics and computer science.
  • Bipartite graphs are fundamental structures with broad applications.

Purpose of the Study:

  • To derive upper bounds for the number of graph homomorphisms from a bipartite graph G to a graph H.
  • To determine upper bounds for the number of dissociation sets within a bipartite graph G.
  • To explore the application of an entropy approach in graph theory.

Main Methods:

  • Utilizing an entropy-based approach.
  • Analyzing properties of graph homomorphisms.
  • Investigating the structure of dissociation sets in bipartite graphs.

Main Results:

  • Established novel upper bounds on the number of graph homomorphisms between bipartite graphs.
  • Provided new upper bounds for the count of dissociation sets in bipartite graphs.
  • Demonstrated the efficacy of the entropy approach in graph theory.

Conclusions:

  • The entropy approach provides a powerful tool for bounding combinatorial quantities in graphs.
  • The derived bounds offer valuable insights into the structure and enumeration of graph homomorphisms and dissociation sets.
  • This research contributes to a deeper understanding of bipartite graph properties.