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Computing degree based topological indices of algebraic hypergraphs.

Amal S Alali1, Esra Öztürk Sözen2, Cihat Abdioğlu3

  • 1Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P. O. Box-84428, Riyadh-11671, Saudi Arabia.

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Summary

This study introduces the prime ideal sum (PIS) hypergraph for commutative rings and computes its topological indices. These indices offer insights into the algebraic structure of rings via hypergraph theory.

Keywords:
05C0705C0905C2505C6513A70Commutative ringHypergraphPrime ideal sum hypergraph(PISH)Topological indicesVertex degree

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Area of Science:

  • Algebraic Graph Theory
  • Commutative Algebra
  • Hypergraph Theory

Background:

  • Topological indices quantify graph and hypergraph topology.
  • Hypergraphs are generalizations of graphs with multi-element edges.
  • Prime ideal sums in commutative rings have significant algebraic properties.

Purpose of the Study:

  • To define a novel hypergraph structure, the prime ideal sum (PIS) hypergraph, for commutative rings.
  • To compute various degree-based topological indices for these PIS hypergraphs.
  • To analyze these indices for specific algebraic structures, including rings of integers modulo n.

Main Methods:

  • Definition of the PIS hypergraph: vertices are non-trivial ideals, edges are maximal subsets of ideals forming a prime ideal.
  • Computation of degree-based topological indices: first and second Zagreb, forgotten, harmonic, Randić, and Sombor indices.
  • Application of methods to specific ring structures, such as Z_n where n is a product of distinct primes.

Main Results:

  • Successful construction of the PIS hypergraph for any commutative ring.
  • Derivation of formulas for several key topological indices applied to PIS hypergraphs.
  • Explicit calculation of these indices for PIS hypergraphs of Z_n for specific n.

Conclusions:

  • The PIS hypergraph provides a new framework for studying algebraic structures using graph-theoretic tools.
  • The computed topological indices offer quantitative measures of the PIS hypergraph's topology, reflecting ring properties.
  • This work bridges abstract algebra and graph theory, opening avenues for further research in algebraic hypergraph theory.