independence fractals of fractal graphs

For an ordered subset $w={w_{1}, w_{2},...,w_{k}}$ of $v(g)$ and a vertex $vin v$, the metric representation of $v$ with respect to $w$ is a $k$-vector, which is defined as $r(v/w)={d(v,w_{1}), d(v,w_{2}),...,d(v,w_{k})}$. the set $w$ is called a resolving set for $g$ if $r(u/w)=r(v/w)$ implies that $u= v$ for all $u,v in v(g)$. the minimum cardinality of a resolving set of $g$ is called the metric dimension of $g$. for two graphs $g$ and $h$, the lexicographic product  $g wr h$ of $h$ by $g$ is obtained from $g$ by replacing each vertex of $g$ with a copy of $h$. a graph $g$ is considered fractal if a graph $gamma$ exists, with at least two vertices, such as $gsimeq gamma wr g$. this paper intends to discuss the fractal graph of some graphs and corresponding independence fractals. also, compare the independent fractals of the fractal graph g, fractal factor $gamma$ and $gamma wr g$.