Score sets in oriented graphs
S. Pirzada, T. A. Naikoo · Applicable Analysis and Discrete Mathematics · 2008
The score of a vertex v in an oriented graph D is , where and are the outdegree and indegree respectively of v and n is the number of vertices in D. The set of distinct scores of the vertices in an oriented graph D is called its score set. If a > 0 and d > 1 are positive integers, we show there exists an oriented graph with score set {a, ad, ad2,..., adn} except for a = 1, d = 2, n > 0, and for a = 1, d = 3, n > 0. It is also shown that there exists no oriented graph with score set {a, ad, ad2,..., adn}, n > 0 when either a = 1, d = 2, or a = 1, d = 3. Also we prove for the non-negative integers a1, a2,..., an with a1 1 i for i=1.