Yazar "Qureshi, Muhammad Imran" seçeneğine göre listele

Listeleniyor 1 - 2 / 2
  • [ X ]
    Öğe
    A graph-theoretic approach to ring analysis: Dominant metric dimensions in zero-divisor graphs
    (Cell Press, 2024) Ali, Nasir; Siddiqui, Hafiz Muhammad Afzal; Riaz, Muhammad Bilal; Qureshi, Muhammad Imran; Akgul, Ali
    This article investigates the concept of dominant metric dimensions in zero divisor graphs (ZDgraphs) associated with rings. Consider a finite commutative ring with unity, denoted as R, where nonzero elements x and y are identified as zero divisors if their product results in zero (x . y = 0). The set of zero divisors in ring R is referred to as L(R). To analyze various algebraic properties of R, a graph known as the zero-divisor graph is constructed using L(R). This manuscript establishes specific general bounds for the dominant metric dimension (Ddim) concerning the ZD-graph of R. To achieve this objective, we examine the zero divisor graphs for specific rings, such as the ring of Gaussian integers modulo m, denoted as Zm[i], the ring of integers modulo n, denoted as Zn, and some quotient polynomial rings. Our research unveils new insights into the structural similarities and differences among commutative rings sharing identical metric dimensions and dominant metric dimensions. Additionally, we present a general result outlining bounds for the dominant metric dimension expressed in terms of the maximum degree, girth, clique number, and diameter of the associated ZD-graphs. Through this exploration, we aim to provide a comprehensive framework for analyzing commutative rings and their associated zero divisor graphs, thereby advancing both theoretical knowledge and practical applications in diverse domains.
  • [ X ]
    Öğe
    Exploring Ring Structures: Multiset Dimension Analysis in Compressed Zero-Divisor Graphs
    (Mdpi, 2024) Ali, Nasir; Siddiqui, Hafiz Muhammad Afzal; Qureshi, Muhammad Imran; Abdallah, Suhad Ali Osman; Almahri, Albandary; Asad, Jihad; Akgul, Ali
    This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring R and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set Z(RE)\{[0]}=RE\{[0],[1]}, where RE={[x] :x is an element of R} and [x]={y is an element of R : ann(x)=ann(y)} is called a compressed zero-divisor graph, denoted by Gamma ER. An edge is formed between two vertices [x] and [y] of Z(RE) if and only if [x][y]=[xy]=[0], that is, iff xy=0. For a ring R, graph G is said to be realizable as Gamma ER if G is isomorphic to Gamma ER. We classify the rings based on Mdim of their associated CZDGs and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Notably, we discuss the interconnection between Mdim, girth, and diameter of CZDGs, elucidating their symmetrical significance.