# Computing degree based topological indices of algebraic hypergraphs

## TL;DR

This paper introduces a new algebraic hypergraph structure based on prime ideal sums in commutative rings and calculates various topological indices for it.

## Contribution

The novel contribution is defining a prime ideal sum hypergraph and computing degree-based topological indices for specific algebraic structures.

## Key findings

- A new hypergraph structure based on prime ideal sums in commutative rings is introduced.
- Several degree-based topological indices are computed for the defined hypergraphs.
- Indices are calculated for specific cases of Zn with different prime factorizations.

## Abstract

Topological indices are numerical parameters that indicate the topology of graphs or hypergraphs. A hypergraph H=(V(H),E(H)) consists of a vertex set V(H) and an edge set E(H), where each edge e∈E(H) is a subset of V(H) with at least two elements. In this paper, our main aim is to introduce a general hypergraph structure for the prime ideal sum (PIS)- graph of a commutative ring. The prime ideal sum hypergraph of a ring R is a hypergraph whose vertices are all non-trivial ideals of R and a subset of vertices Ei with at least two elements is a hyperedge whenever I+J is a prime ideal of R for each non-trivial ideal I, J in Ei and Ei is maximal with respect to this property. Moreover, we also compute some degree-based topological indices (first and second Zagreb indices, forgotten topological index, harmonic index, Randić index, Sombor index) for these hypergraphs. In particular, we describe some degree-based topological indices for the newly defined algebraic hypergraph based on prime ideal sum for Zn where n=pα,pq,p2q,p2q2,pqr,p3q, p2qr,pqrs for the distinct primes p,q,r and s.