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Degree sums and dense spanning trees.

Tao Li1, Yingqi Gao1, Qiankun Dong1

  • 1College of Computer and Control Engineering, Nankai University, Tianjin 300071, P.R. China.

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Summary
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This study explores finding dense spanning trees (DST) in graphs, a variant of the minimum spanning tree (MST) problem. Researchers developed conditions based on vertex degrees to identify dense spanning subtrees, offering insights for related algorithms.

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Area of Science:

  • Graph theory
  • Discrete mathematics
  • Computer science

Background:

  • The minimum spanning tree (MST) problem is well-studied in graph theory.
  • Finding dense spanning trees (DST) is a related but less explored problem.
  • Existing methods lack efficient approaches for identifying dense structures.

Purpose of the Study:

  • To investigate conditions for finding dense spanning trees (DST) in unweighted graphs.
  • To adapt mathematical properties of extremal structures for DST.
  • To provide foundational insights for developing efficient DST algorithms.

Main Methods:

  • Utilized mathematical properties of extremal structures with minimum vertex distance sums.
  • Formulated general conditions based on the sum of vertex degrees.
  • Analyzed the performance of degree sum combinations for finding dense spanning subtrees.
  • Applied the approach to practical examples and DST variations.

Main Results:

  • Established general conditions on vertex degree sums for identifying dense spanning trees.
  • Demonstrated the application of these conditions in finding dense spanning subtrees.
  • Showcased the algorithm's adaptability to variations of DST and MST problems.
  • Gained insights into the role of degree sums in forming dense spanning trees.

Conclusions:

  • The study provides a novel approach to finding dense spanning trees using vertex degree conditions.
  • The developed conditions offer a foundation for future research in efficient DST algorithms and heuristics.
  • This work contributes to understanding the structural properties relevant to dense graph traversals.