Some Properties of Graphs with Multiple Edges

Abstract

In this paper we consider undirected graphs, with no edges joining a vertex to itself, but with possibly several edges joining pairs of vertices. The first part of the paper deals with the question of characterizing those sets of non-negative integers d₁d₂ . . . , d_n and {c_ij}, 1 ≤ i < j ≤ n, such that there exists a graph G with n vertices whose valences (degrees) are the numbers d_i, and with the additional property that the number of edges joining i and j is at most c_ij. This problem has been studied extensively, in the general case (1, 2, 9, 11), in the case where the graph is bipartite (3, 5, 7, 10), and in the case where the C_ij are all 1 (6).