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In this paper we use number-theoretic properties to classify ordinary graphs that are finite and have no isolated vertices. The classification depends on whether there is an assignment of real values, usually rational integer values, to the edges of the graph, such that the set of assigned values and the set of vertex sums of these values, summed at each vertex over all the edges incident to the vertex, will be a pair of sets with prescribed properties. Then we seek corresponding graph-theoretic properties.It is possible to describe the problem in terms of a symmetric matrix having specified properties for its row sums, but in this paper we make no use of this interpretation; however, see (3).

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An ordinary graph is a finite linear graph which contains no loops or multiple edges, and in which all edges are undirected. In such a graph G, let N, L, and T denote respectively the number of nodes, edges, and triangles. One problem, suggested by P. Erdös (1), is to determine the minimum number of triangles when the number of edges is specified, subject to suitable restrictions.

In our paper “Magic graphs” (1) we showed that every complete graph Kn with n ⩾ 5 is “magic,” i.e., if the vertex set is indicated {vi} and if eij is the edge joining vi and vj, i ≠ j , then there exists a function α(eij) such that the set {α(eij)} consists of distinct positive rational integers and the vertex sums1have a constant value σ(α) for k = 1, 2, … , n. We noted that K2 is magic and showed that K3 and K4 are not magic.