Generalized sum graphs

Harary [8] calls a finite, simple graph G a sum graph if one can assign to each vi ∈ V (G) a label xi so that {vi, vj} ∈ E(G) iff xi + xj = xk for some k. We generalize this notion by replacing “xi + xj” with an arbitrary symmetric polynomial f(xi, xj). We show that for each f , not all graphs are “f -graphs”. Furthermore, we prove that for every f and every graph G we can transform G into an f -graph via the addition of |E(G)| isolated vertices. This result is nearly best possible in the sense that for all f and for all m ≤ 1 2 ( n 2 ) , there is a graph G with n vertices and m edges which, even after the addition of m−O(n log n) isolated vertices, is not an f -graph. ∗Research supported in part by a U.S.A.-Israel Binational Science Foundation and by a Bergmann Memorial Grant. †Research supported in part by the Office of Naval Research contract number N00014– 85–K0622.