Computing degree based topological indices of algebraic hypergraphs

Topological indices are numerical parameters that indicate the topology of graphs or hypergraphs. A hypergraph $H=\left(V\right(H),E(H\left)\right)$ consists of a vertex set $V\left(H\right)$ and an edge set $E\left(H\right)$ , where each edge $e\in E\left(H\right)$ is a subset of $V\left(H\right)$ 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 $E_i$ 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 $E_i$ and $E_i$ 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 $Z_n$ where $n=p^{\alpha },pq,p^{2}q,p^2{q}^2,pqr,p^{3}q$ , $p^{2}qr,pqrs$ for the distinct primes $p,q,r$ and s.