Document Type : Research Paper

Authors

• Shahida A T ¹
• Minirani S ²
• Sreeji P C ¹

¹ Department of Mathematics, M E S Mampad College, Malappuram, India

² MPSTME, NMIMS University Mumbai, Mumbai, India

Abstract

For an ordered subset $W=\{w_1,w_2,...,w_k\}$ 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 $r(v/W)=\{d(v,w_1),d(v,w_2),...,d(v,w_k)\}$. 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 $\mathrm{\Gamma }$ exists, with at least two vertices, such as $G\simeq \mathrm{\Gamma }\wr G$. 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 $\mathrm{\Gamma }$ and $\mathrm{\Gamma }\wr G$.

Keywords

• Fractal graph
• Egamorphism
• Metric dimension
• Metric basis
• Resolving set
• Independence Fractals