The king degree and the second out-degree of tournaments

Abstract

In a digraph, the second out-degree of a vertex x, denoted by $d^{++}\left(x\right)$, is the number of vertices y such that $d^{+}(x,y)=2$, where $d^{+}(x,y)$ is the length of the shortest xy-directed path, if it exists. It is obvious that the sum of the first out-degrees of the vertices in a digraph is nothing but the number of its arcs. Unlike the first out-degree, the summation of the second out-degrees of the vertices in a digraph is not constant with respect to the number of vertices and arcs. In this paper, we characterize, as a function of some integer n, the values that can be the summation of the second out-degrees of the vertices in a tournament of order n. Throughout the paper, we use the new concept of king degree in order to settle the problem. The king degree of a vertex x is the number of vertices that can be reached from x by a directed path of length at most 2. Several open problems are introduced in the last section of the paper.