On Balance Index Sets of One-Point Unions 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 an edge partial labeling $f^{\ast }:E(G)\to A$ defined by $f^{\ast }(xy)=f(x)$ if and only if $f(x)=f(y)$ for each edge $xy\in E(G)$. For each $i\in A$, 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 $BI(G)$, is defined as

$$\{|e_f(0)–e_f(1)|:|v_f(0)–v_f(1)|\le 1\}.$$

In this paper, exact values of the balance index sets of five new families of one-point union of graphs are obtained, many of them, but not all, form arithmetic progressions.