Muhammad SALMAN, Imran JAVAID, Muhammad Anwar CHAUDHRY
A set
W of the vertices of a connected graph
G is called a resolving set for
G if for every two distinct vertices
u, v ∈
V (
G) there is a vertex
w ∈
W such that
d(
u,w)≠
d(
v,w). A resolving set of minimum cardinality is called a metric basis for
G and the number of vertices in a metric basis is called the metric dimension of
G, denoted by dim(
G). For a vertex
u of
G and a subset
S of
V (
G), the distance between
u and
S is the number min
s∈S d(
u, s). A
k-partition Π={
S1,
S2,...,
Sk} of
V (
G) is called a resolving partition if for every two distinct vertices
u, v ∈
V (
G) there is a set
Si in Π such that
d(
u,
Si)≠
d(
v,
Si). The minimum
k for which there is a resolving
k-partition of
V (
G) is called the partition dimension of
G, denoted by pd(
G). The circulant graph is a graph with vertex set Z
n, an additive group of integers modulo
n, and two vertices labeled
em and
j adjacent if and only if
i?j (mod
n)∈
C, where
C⊂Z
n has the property that
C=-
C and 0∉C. The circulant graph is denoted by
Xn,Δ where Δ=|
C|. In this paper, we study the metric dimension of a family of circulant graphs
Xn,3 with connection set
C={1,
n/2,
n-1} and prove that dim(
Xn,3) is independent of choice of
n by showing that
We also study the partition dimension of a family of circulant graphs
Xn,4 with connection set
C={±1,±2} and prove that pd(
Xn,4) is independent of choice of
n and show that pd(
X5,4)=5 and
.
