WIENER AND HARARY INDICES OF MYCIELSKIAN GRAPHS

Year 2025, Volume: 15 Issue: 11, 2687 - 2696, 03.11.2025

Shanu Goyal Tanya Mittal

Abstract

Let $G = (V(G), E(G))$ be a graph, where $V= \{v_{1}, v_{2}, \ldots v_{n} \}$. Let $V' = \{ v'_{1}, v'_{2}, \ldots, v'_{n} \}$ be the twin of the vertex set $V(G)$. The Mycielskian graph $mathbb{M}(G)$ of $G$ is defined as the graph whose vertex set is $V(G) \cup V'(G) \cup \{w\}$ and the edge set is $E(G) \cup \{v_{i}v'_{j}: v_{i}v_{j}\in E(G) \} \cup \{v'_{i}w \in V'(G) \}$. The vertex $v'_{i}$ is the twin of the vertex $v_{i}$ (or $v_{i} $ is twin of the vertex $v'_{i}$) and the vertex $w$ is the root of $\mathbb{M}(G)$. The closed Mycielskian graph $\mathbb{M}[G]$ of $G$ is defined as the graph whose vertex set is $V(G) \cup V'(G) \cup \{w\}$ and the edge set is $E(G) \cup \{v_{i}v'_{j}: v_{i}v_{j}\in E(G) \} \cup\{v_{i}v'_{i}: i = 1, 2, \ldots, n \} \cup\{v'_{i}w \in V'(G) \}$. The vertex $v'_{i}$ is the twin of the vertex $v_{i}$ (or $v_{i} $ is twin of the vertex $v'_{i}$) and the vertex $w$ is the root of $\mathbb{M}[G]$.
In this paper, we study the Wiener and Harary indices of the Mycielskian and closed Mycielskian graphs.

Keywords

closed splitting graph , shadow graph , closed shadow graph , Mycielskian graph , closed Mycielskian graph

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