Generalized sum graphs

AbstractHarary [8] calls a finite, simple graphG asum graph if one can assign to eachvi∈V(G) a labelxi so that{vi,vj}∈E(G) iffxi+xj=xk for somek. We generalize this notion by replacing “xi+xj” with an arbitrary symmetric polynomialf(xi,xj). We show that for eachf, not all graphs are “f-graphs”. Furthermore, we prove that for everyf and every graphG we can transformG into anf-graph via the addition of |E(G)| isolated vertices. This result is nearly best possible in the sense that for allf and for $$ \leqslant \frac{1}{2}\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)$$ , there is a graphG withn vertices andm edges which, even after the addition ofm-O(n logn) isolated vetices, is not andf-graph.