A NOTE ON SIGN BALANCED INDEX SET OF A GRAPH

Let $G$ be a graph with vertex set $V$ and edge set $E$. Let $g$ be a labeling from $E$ to $\{+, -\}$. The edge labeling $g$ induces a vertex labeling $h: V\rightarrow \{+, -\}$ defined by $h(v)=\prod g(uv)$ for $u$ in $N(v)$, where $N(v)$ is the set of vertices adjacent to $v$. Let $e(+)=card\{e \in E :g(e)= +\}$, $e(-)=card\{e \in E:g(e)= - \}$ and $v(+)=card\{v \in V : h(v)= + \}$, $v(-)=card\{v \in V : h(v)= -\}$. A labeling $g$ is said to be sign friendly if $\mid e(+) - e(-) \mid \leq 1$. The sign balanced index set $(SBIS)$ of a graph $G$ is defined by $\{\mid v(+) - v(-) \mid$ : the edge labeling $g$ is sign friendly\}. In this paper we completely determine the sign balanced index sets of some important family of graphs.