On Balance Index Sets of Halin graphs of Stars and Double Stars

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 $f^{\ast }((u,v))=f(u)$ if and only if $f(u)=f(v)$ for each edge $(u,v)\in E(G)$. For $i\in A$, let ${\text{v}}_f(i)=\text{card}\{v\in V(G):f(v)=i\}$ and ${\text{e}}_f(i)=\text{card}\{e\in E(G):f^{\ast }(e)=i\}$. A labeling $f$ of $G$ is said to be friendly if $|{\text{v}}_f(0)–{\text{v}}_f(1)|\le 1$. The balance index set of the graph $G$, $\text{BI}(G)$, is defined as $\{|{\text{e}}_f(0)–{\text{e}}_f(1)|:\text{the vertex labeling }f\text{ is friendly}\}$. We determine the balance index sets of Halin graphs of stars and double stars.