Computing Metric Dimension of Power of Total Graph

For a connected graph
$\mathcal {G}$
, the distance
$d(u, v)$
between any two vertices
$u$
,
$v~\in ~\mathcal {V(G)}$
is defined as;
$d(u,v)=\min _{\mathcal {P}_{u,v}}\{\text {length of}~\mathcal {P}_{uv}\}$
i.e the minimum length of path connecting any two vertices. Two vertices
$u$
,
$v$
of a graph
$G$
are said to be resolved by a vertices
$\mathcal {W}$
of a graph
$\mathcal {G}$
if
$d(u,w) \neq d(v,w)$
. An ordered set
$\mathcal {W} = \{w_{1},w_{2},w_{3},\ldots,w_{k}\}\,\,\in \,\,V(G)$
is resolving set for a connected graph
$G$
, if for any two vertices
$u$
,
$v$
there exists
$w_{i} \in W$
such that
$d(u,w_{i}) \neq d(v, w_{i})$
. The representation of vertices
$v$
w.r.t
$\mathcal {W}$
is represented by
$r(v|\mathcal {W})$
which is
$k$
-vector(
$k$
-tuple)
$(d(v,w_{1}),d(v,w_{2}),d(v,w_{3}),\ldots,d(v,w_{k}))$
. If the representation of all the vertices of graph
$\mathcal {G}$
are different w.r.t
$\mathcal {W}$
then
$\mathcal {W}$
is the resolving set for
$G$

[1]
. A resolving set that contains minimum numbers of vertices is
metric basis
for
$\mathcal {G}$
. The cardinality of smallest resolving set is the
metric dimension
of
$\mathcal {G}$
, represented by
$dim(\mathcal {G})$

[2]
,
[3]
. A family of connected graph
$\mathcal {G}$
has unbounded metric dimension if
$dim(\mathcal {G})$
depends on order of a graph. In this paper, we show that total graph of path power three and four have unbounded metric dimension. We also proved some results on edges of power of path graph.