Harary's Sum and Integral Sum Graphs -- A Survey in detail

Abstract

Harary introduced the concepts of sum and integral sum graphs. A graph $G$ is a {\em sum graph} if the vertices of $G$ can be labeled with distinct positive 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 $G$. An {\em integral sum graph} is also defined just as sum graph, the difference being that the labels may be any distinct integers. In this survey paper, we present results obtained on sum and integral sum graphs by different authors in detail.