Computing the partition dimension of certain families of Toeplitz graph - Luo, Ricai; Khalil, Adnan; Ahmad, Ali; Azeem, Muhammad; Ibragimov, Gafurjan; Nadeem, Muhammad Faisal (Frontiers in Computational Neuroscience, 2022)
Let $ G=(V(G),E(G))$ be a graph with no loops, numerous edges, and only one component, which is made up of the vertex set $V(G)$ and the edge set $E(G).$ The distance $d(u,v)$ between two vertices $u,v$ that belong to the vertex set of $H$ is the shortest path between them. A $k$-ordered partition of vertices is defined as $β=\{β_1,β_2,…,β_k\}.$ If all distances $d(v,β_k)$ are finite for all vertices $v\in V,$ then the $k$-tuple $(d(v,β_1),d(v, β_2),…, d(v, β_k))$ represents vertex $v$ in terms of $β,$ and is represented by $r(v|β).$ If every vertex has a different presentation, the $k$-partition $β$ is a resolving partition. The partition dimension of G, indicated by $pd(G),$ is the minimal $k$ for which there is a resolving $k$-partition of $V(G).$ The partition dimension of Toeplitz graphs formed by two and three generators is constant, as shown in the following paper. Resolving set allows to obtain 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 represents the atom and bond types, respectively.