Generalized sum graphs

Abstract

Harary [8] calls a finite, simple graphG asum graph if one can assign to eachv _i ∈V(G) a labelx _i so that{v _i ,v _j }∈E(G) iffx _i +x _j =x _k for somek. We generalize this notion by replacing “x _i +x _j ” with an arbitrary symmetric polynomialf(x _i ,x _j ). 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.