Computing the partition dimension of certain families of Toeplitz graph

Abstract

LetG= (V(G),E(G)) be a graph with no loops, numerous edges, and only one component, which is made up of the vertex setV(G) and the edge setE(G). The distanced(u, v) between two verticesu, vthat belong to the vertex set ofHis the shortest path between them. Ak-ordered partition of vertices is defined as β = {β1, β2, …, βk}. If all distancesd(v, βk) are finite for all verticesv∈V, then thek-tuple (d(v, β1),d(v, β2), …,d(v, βk)) represents vertexvin terms of β, and is represented byr(v|β). If every vertex has a different presentation, thek-partition β is a resolving partition. The partition dimension of G, indicated bypd(G), is the minimalkfor which there is a resolvingk-partition ofV(G). The partition dimension of Toeplitz graphs formed by two and three generators is constant, as shown in the following paper. The resolving set allows obtaining a unique representation for computer structures. In particular, they are used in pharmaceutical research for discovering patterns common to a variety of drugs. The above definitions are based on the hypothesis of chemical graph theory and it is a customary depiction of chemical compounds in form of graph structures, where the node and edge represent the atom and bond types, respectively.