Magic Graphs

Abstract

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).