Abstract
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
∈
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. 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.