Independence fractals of fractal graphs

Document Type : Research Paper

Authors

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 $\Gamma$ 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

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Volume 14, Issue 10October 2023Pages 239-246
• Receive Date: 05 May 2022
• Revise Date: 16 June 2022
• Accept Date: 08 July 2022