Some Results on Sum Graph, Integral Sum Graph and Mod Sum Graph

Abstract:

      Let $N$ denote the set of positive integers. The sum graph $G^{+}(S)$ of a finite subset $S\subset N$ is the graph $(S,E)$ with $uv\in E$ if and only if $u+v\in S$. A graph $G$ is said to be a sum graph if it is isomorphic to the sum graph of some $S\subset N$. By using the set $Z$ of all integers instead of $N$, we obtain the definition of the integral sum graph. A graph $G=(V,E)$ is a mod sum graph if there exists a positive integer $z$ and a labelling, $\lambda$, of the vertices of $G$ with distinct elements from $\{0,1,2,\ldots,z-1\}$ so that $uv\in E$ if and only if the sum, modulo $z$, of the labels assigned to $u$ and $v$ is the label of a vertex of $G$. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch $m$-wind-mill ($D_{m}$) is integral sum graph and mod sum graph, and give the sum number of $D_{m}$.