Abstract A digraph D is called quasi-transitive if, for every triple x, y, z of distinct vertices of
D such that xy and yz are arcs of D, there is at least one are between x and z. Seymour’s second
neighbourhood conjecture asserts that every oriented graph D has a vertex v such that d+D(x) 6 d++D (x),
where an oriented graph is a digraph with no cycle of length two. A vertex that satisfies Seymour’s
second neighbourhood conjecture is called a Seymour vertex. Fisher proved that Seymour’s second
neighbourhood conjecture restricted to tournaments is true, where any tournament contains at least
one Seymour vertex. Havet and Thomass′e proved that a tournament T with no vertex of out-degree
zero has at least two Seymour vertices. Observe that quasi-transitive oriented graphs is a superclass of
tournaments. In this paper, Seymour’s second neighbourhood conjecture on quasi-transitive oriented
graphs is the core problem. Notice the relationship between Seymour vertices of a quasi-transitive
oriented graph and an extended tournament. It is proved that the conjecture is true for quasi-transitive
oriented graphs. Furthermore, every quasi-transitive oriented graph has at least a Seymour vertex and
every quasi-transitive oriented graph with no vertex of out-degree zero has at least two Seymour vertices.
LI Rui-juan, SHI Jie, ZHANG Xin-hong. Seymour vertices in a quasi-transitive oriented graph. Applied Mathematics A Journal of Chinese Universities, 2020, 35(2): 245-252.
HOU Yuan, CHEN Yu-li, ZHENG Yi-rong. Hyper-Wiener index of Eulerian graphs[J]. Applied Mathematics A Journal of Chinese Universities, 2016, 31(2): 248-252.