Independence fractals of fractal graphs

For an ordered subset W=\{w_{۱}, w_{۲},...,w_{k}\} of V(G) and a vertex v\in V, the metric representation of v with respect to W is a k-vector, which is defined as r(v/W)=\{d(v,w_{۱}), d(v,w_{۲}),...,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 G\simeq \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.

Keywords:

Fractal graph , Egamorphism , Metric Dimension , Metric basis , Resolving set , Independence Fractals