Let $n\ge 1$ be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of ${\mathbb{Z}}_n$, where $n=p^{\alpha },p^{2}q,p^2{q}^2,pqr,p^{3}q,$$p^{2}qr$, and $pqrs$ for the different prime integers $p,q,r,$ and s. Moreover, we construct M-polynomials and $CoM$-polynomials using the $PIS$-graph structure of ${\mathbb{Z}}_n$ to avoid the difficulty of computing the descriptors via formulas directly. Furthermore, we present a geometric comparison for representations of each surface obtained by M-polynomials and $CoM$-polynomials. Finally, we discuss the applicability of algebraic graphs to chemical graph theory.

In this manuscript, we interpret the theory of bipolar complex fuzzy submodule (BCFSM) of a provided classical module over a ring by utilizing the well-known notion of bipolar complex fuzzy set (BCFS). We also devise the sum of two BCFS and investigate its related proposition for studying a few fundamental properties of the initiated BCFSM. Moreover, we investigate the cartesian product of two BCFS for developing properties of BCFSM. We also investigate that if a BCFS Ӈ 1 is a BCFSM of Ӎ 1 , then image Ӻ Ӈ is a BCFSM of Ӎ 2 and the preimage Ӻ − 1 Ӈ is a BCFSM of Ӎ 1 , where, Ӎ 1 and Ӎ 2 are two modules and Ӻ : Ӎ 1 → Ӎ 2 is a module homomorphism.

A Fermatean fuzzy set (FRFS) is the extension of a fuzzy set, an intuitionistic fuzzy set, and a Pythagorean fuzzy set, and is used in different fields. Unlike other fuzzy structures, the sum of cubes of membership grades in FRFSs approximates a unit interval, increasing uncertainty. In this study, we intend to provide unique operational rules and aggregation operators (AOs) inside a Fermatean fuzzy environment. To develop a fair remedy for the membership degree and non-membership degree features of “Fermatean fuzzy numbers (FRFNs)”, our solution introduces new neutral or fair operating principles, which include the concept of proportional distribution. Based on the suggested operating principles, we provide the “Fermatean fuzzy fairly weighted average operator and the Fermatean fuzzy fairly ordered weighted averaging operator”. Our suggested AOs provide more generalized, reliable, and exact data than previous techniques. Combining the recommended AOs with multiple decision-makers and partial weight information under FRFSs, we also devised a technique for “multi-criteria decision-making”. To illustrate the application of our novel method, we provide an example of the algorithm’s effectiveness in addressing decision-making challenges.

A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation are central, then the ring is commutative. In the present paper, we extended this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we established the ∗-version of Vukman’s theorem. Precisely, we describe the structure of quotient ring $\mathfrak{A}/\mathfrak{L}$, where $\mathfrak{A}$ is an arbitrary ring and $\mathfrak{L}$ is a prime ideal of $\mathfrak{A}$. Further, by taking advantage of the ∗-version of Vukman’s theorem, we show that if a 2-torsion free semiprime $\mathfrak{A}$ with involution admits a nonzero ∗-centralizing derivation, then $\mathfrak{A}$ contains a nonzero central ideal. This result is in the spirit of the classical result due to Bell and Martindale (Theorem 3). As the applications, we extended and unified several classical theorems. Finally, we conclude our paper with a direction for further research.

In 1997, Sierpinski graphs, $S(n,k)$, were obtained by Klavzar and Milutinovic. The graph $S(1,k)$ represents the complete graph $K_k$ and $S(n,3)$ is known as the graph of the Tower of Hanoi. Through generalizing the notion of a Sierpinski graph, a graph named a generalized Sierpinski graph, denoted by $Sie(\Lambda ,t)$, already exists in the literature. For every graph, numerous polynomials are being studied, such as chromatic polynomials, matching polynomials, independence polynomials, and the M-polynomial. For every polynomial there is an underlying geometrical object which extracts everything that is hidden in a polynomial of a common framework. Now, we describe the steps by which we complete our task. In the first step, we generate an M-polynomial for a generalized Sierpinski graph $Sie(\Lambda ,t)$. In the second step, we extract some degree-based indices of a generalized Sierpinski graph $Sie(\Lambda ,t)$ using the M-polynomial generated in step 1. In step 3, we generate the entropy of a generalized Sierpinski graph $Sie(\Lambda ,t)$ by using the Randić index.