SCORE SEQUENCES IN ORIENTED GRAPHS

An oriented graph is a digraph with no symmetric pairs of directed arcs and without loops. The score of a vertex $v_i$ in an oriented graph D is $a_{v_i}\;(or\;simply\;a_i)=n-1+d_{v_i}^+-d_{v_i}^-,\;where\; d_{v_i}^+\;and\;d_{v_i}^-$ are the outdegree and indegree, respectively, of $v_i$ and n is the number of vertices in D. In this paper, we give a new proof of Avery's theorem and obtain some stronger inequalities for scores in oriented graphs. We also characterize strongly transitive oriented graphs.