On Balance Index Sets of Generalized Book and Ear Expansion Graphs

Abstract

Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$, and let ${\mathbb{Z}}_2=\{0,1\}$. A labeling $f:V(G)\to {\mathbb{Z}}_2$ induces a partial edge labeling $f^{\ast }:E(G)\to {\mathbb{Z}}_2$ defined by $f^{\ast }(uv)=f(u)$ if and only if $f(u)=f(v)$. For $i\in {\mathbb{Z}}_2$, let $V_f(i)=\{v\in V(G):f(v)=i\}$ and $e_f(i)=|\{e\in E(G):f^{\ast }(e)=i\}|$. A labeling $f$ is called a friendly labeling if $|V_f(0)–V_f(1)|\le 1$. The $BI(G)$, the balance index set of $G$, is defined as $\{|e_f(0)–e_f(1)|:\text{the vertex labeling }f\text{ is friendly}\}$. This paper focuses on the balance index sets of generalized book and ear expansion graphs.