On The Balance Index Sets of Graphs

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $A=\{0,1\}$. A labeling $f:V(G)\to A$ induces a partial edge labeling $f^{\ast }:E(G)\to A$ defined by $f^{\ast }(xy)=f(x),\text{ if and only if }f(x)=f(y),$ for each edge $xy\in E(G)$. For $i\in A$, let $v_f(i)=\text{card}\{v\in V(G):f(v)=i\}$ and $e_f^{\ast }(i)=\text{card}\{e\in E(G):f^{\ast }(e)=i\}.$ A labeling $f$ of a graph $G$ is said to be friendly if $|v_f(0)–v_f(1)|\le 1.$ If $|e_f(0)–e_f(1)|\le 1,$then $G$ is said to be \textbf{\emph{balanced}}. The \textbf{\emph{balance index set}} of the graph $G$, $BI(G)$, is defined as $BI(G)=\{|e_f(0)–e_f(1)|:\text{the vertex labeling }f\text{ is friendly}\}.$Results parallel to the concept of friendly index sets are pr