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Minimum-Loop Realization of Degree Sequences.

A J Goldman1, R H Byrd1

  • 1The Johns Hopkins University, Baltimore, MD 21218.

Journal of Research of the National Bureau of Standards (1977)
|September 27, 2021
PubMed
Summary
This summary is machine-generated.

The minimum number of loops required for a graph realization of a degree sequence D is determined by its maximum element and sum. This research provides insights into graph theory and loop minimization problems.

Keywords:
degree sequencegraphincidence sequenceloopless graphpartition

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Area of Science:

  • Graph Theory
  • Combinatorics
  • Discrete Mathematics

Background:

  • Degree sequences characterize graphs.
  • Conditions for graph realization (even sum S(D)) and loopless graph realization (S(D) - 2M(D) >= 0) are established.
  • M(D) denotes the maximum element and S(D) the sum of a degree sequence D.

Purpose of the Study:

  • To determine the minimum number of loops required for graph realizations of degree sequences.
  • To investigate the uniqueness of minimum-loop realizations.
  • To extend these findings to a generalized loop-cost minimization problem.

Main Methods:

  • Analysis of the relationship between degree sequence properties (M(D), S(D)) and graph structure.
  • Derivation of a formula for the minimum number of loops based on the expression 2M(D) - S(D).
  • Extension of results to weighted loop costs.

Main Results:

  • If 2M(D) - S(D) is positive and even, then (2M(D) - S(D))/2 is the minimum number of loops for a graph realizing D.
  • The minimum-loop realization of D is unique.
  • The set of possible numbers of loops in graphs realizing D is determined.

Conclusions:

  • The study provides a precise method for calculating the minimum number of loops in graph realizations of degree sequences.
  • The findings contribute to understanding the structure of graphs with specific degree sequences.
  • The research offers solutions for generalized loop-cost minimization problems in graph theory.