Generalized sum graphs (Q1187945)

This paper deals with \(f\)-graphs defined in the following way. Given a symmetric polynomial of two variables \(f:\mathbb{R}^ 2\to\mathbb{R}\). Then a graph \(G\) is said to be an \(f\)-graph if one can assign real numbers \(x_ 1,x_ 2,\dots,x_ n\) to its vertices \(v_ 1,v_ 2,\dots,v_ n\) so that \(\{v_ i,v_ j\}\in E(G)\) iff \(f(x_ i,x_ j)=x_ k\) for some \(k\). The main result of this paper is that for any symmetric polynomial \(f\) not all graphs are \(f\)-graphs and that for every graph \(G\) there is a symmetric polynomial \(f\) with the property that \(G\) is an \(f\)-graph.