Edge Sum Index of a Graph in a Commutative Ring

Abstract

Let $\Gamma(Z_{n})$ be a graph. A bijection $f:E(\Gamma(Z_{n})) \to Z^{+}$, where $Z^{+}$ is a set of positive integers is called an edge mapping of the graph $\Gamma(Z_{n})$. Now, we define, $F(v)=\Sigma\{f(e);$ e is incident on $v \}$ on $V(\Gamma(Z_{n}))$. Then, F is called the edge sum mapping of the edge mapping f. $\Gamma(Z_{n})$ is said to be an edge sum graph if there exists an edge mapping $f:E(\Gamma(Z_{n})) \to N^{+}$ such that f and its corresponding edge sum mapping. F on $V(\Gamma(Z_{n}))$ satisfy the following conditions: (i) F is into mapping to$ Z^{+}$. That is, $F(v) \in Z^{+}$, for every $v \in E(\Gamma(Z_{n}))$. (ii) If $e_{1},e_{2},\dots,e_{n} \in E(\Gamma(Z_{n}))$ such that $f(e_{1})+f(e_{2})+\dots f(e_{n}) \in Z^{+}$, then $e_{1},e_{2},\dots,e_{n}$ are incident on a vertex in $\Gamma(Z_{n})$. In this paper, we evaluated the edge sum index of some standard graphs in zero divisor graph.