On Balance Index Sets of Generalized Wheels

Abstract

A vertex labeling $f:V\to \{0,1\}$ of the simple graph $G=(V,E)$ induces a partial edge labeling $f^{\ast }:E\to \{0,1\}$ defined by $f^{\ast }(uv)=f(u)$ if and only if $f(u)=f(v)$. Let $v(i)$ and $e(i)$ be the number of vertices and edges, respectively, that are labeled $i$, and define the balance index set of $G$ as $\{|e(0)–e(1)|:|v(0)–v(1)|\le 1\}$. In this paper, we determine the balance index sets of generalized wheels, which are the Zykov sum of a cycle with a null graph.