Classification of posets using zero-divisor graphs

Abstract

Halaš and Jukl associated the zero-divisor graph G to a poset (X,≤) with zero by declaring two distinct elements x and y of X to be adjacent if and only if there is no non-zero lower bound for {x, y}. We characterize all the graphs that can be realized as the zero-divisor graph of a poset. Using this, we classify posets whose zero-divisor graphs are the same. In particular we show that if V is an n-element set, then there exist $\begin{array}{} \sum\limits_{\log_2(n+1)\leq k\leq n}^{}\binom{n}{k}\binom{2^k-k-1}{n-k} \end{array} $ reduced zero-divisor graphs whose vertex sets are V.

On the metric determination of linear dependence graph

“…Beck 1 was the pioneer to initiate the study on this topic when he determined a graph associated with a commutative ring by means of coloring. Later on, this topic gained the attention of many researchers which resulted a number of remarkable determinations of various types of graphs associated with groups, 2‐12 rings, 13‐15 ideals, 16‐18 vector spaces, 19‐24 and posets 25‐28 …”

“…Later on, this topic gained the attention of many researchers which resulted a number of remarkable determinations of various types of graphs associated with groups, [2][3][4][5][6][7][8][9][10][11][12] rings, [13][14][15] ideals, [16][17][18] vector spaces, [19][20][21][22][23][24] and posets. [25][26][27][28] The problem of vertex determination in a connected graph G by defining the metric on G attracted various graph theorist as well many other researchers due to its significant applications in several extents including verification, security and discovery of networks, 29 the chemistry of pharmaceutics for drug designing, 30 mastermind game strategies, 31 navigation of robots, 32 connected joins in graphs, 33 and solution of coin weighing problems. 34 Because of these practical significance of this problem, from the last two decades, numerous researchers determined vertices in various families of graphs, mainly, by defining metric related concepts of the metric dimension, 30,35,36 and then by defining several variations of the metric dimension.…”

On the spectrum of linear dependence graph of a finite dimensional vector space

“…This new technique of studying algebraic structures leads to many fascinating results and questions. For various constructions of graphs on different algebras we refer to [2,7] on rings, [3,5] on groups, [6] on semigroups, [23,24,26,33] on posets, [9,10,11,12] on vector spaces.…”