On Balance Index Sets of Trees of Diameter Four

Abstract

Let $G$ be a simple graph with a vertex set $V(G)$ and an edge set $E(G)$, and let ${\mathbb{Z}}_2=\{0,1\}$. A labeling $f:V(G)\to {\mathbb{Z}}_2$ induces an edge partial labeling $f^{\ast }:E(G)\to {\mathbb{Z}}_2$ defined by $f^{\ast }(xy)=f(x)$ if and only if $f(x)=f(y)$ for each edge $xy\in E(G)$. For each $i\in {\mathbb{Z}}_2$, let $v_f(i)=|\{v\in V(G):f(v)=i\}|$ and $e_f(i)=|\{e\in E(G):f^{\ast }(e)=i\}|$. The balance index set of $G$, denoted $\text{BI}(G)$, is defined as $\{|e_f(0)–e_f(1)|:|v_f(0)–v_f(1)|\le 1\}$. In this paper, we investigate and present results concerning the balance index sets of trees of diameter four.