Banana Trees and Union of Stars are Integral Sum Graphs

Abstract

The concept of integral sum graphs is introduced by Harary $[6]$. A graph $G$ is an integral sum graph or $\int \mathrm{\Sigma }$-graph if the vertices of $G$ can be labelled with distinct integers so that e = uv is an edge of G if and only if the sum of the labels on vertices $u$ and $v$ is also a label in G. Xu $[12]$ has shown that the union of any three stars and the union of any number of integral sum trees are integral sum graphs. Xu poses the question as to whether all disconnected forests are integral sum graphs. In this paper, we prove that all banana trees and union of any number of stars are integral sum graphs.