An n × n complex matrix A is h-pseudo-tournament if A + A∗ = hh∗ − I, where h is a complex, non-zero column vector. The class of h-pseudo-tournament matrices is a generalization of the well studied tournament-like matrices: h-hypertournament matrices, generalized tournament matrices and tournament matrices. In this paper we derive new spectral properties of an h-pseudo-tournament matrix. When t...

If x is a vertex of a digraph D, then we denote by d(x) and d−(x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D) = max{d+(x), d−(x)}−min{d+(y), d−(y)} over all vertices x and y of D (including x = y). If ig(D) = 0, then D is regular and if ig(D) ≤ 1, then D is almost regular. A c-partite tournament is an orientation of a complete c-...

We show that every regular tournament on n vertices has at least n!/(2+o(1)) Hamiltonian cycles, thus answering a question of Thomassen [17] and providing a partial answer to a question of Friedgut and Kahn [7]. This compares to an upper bound of about O(nn!/2) for arbitrary tournaments due to Friedgut and Kahn (somewhat improving Alon’s bound of O(nn!/2)). A key ingredient of the proof is a ma...

Let e(Tn) be the primitive exponent of a primitive tournament Tn of order n. In this paper, we obtain the following results. 1. Let Tn be a regular or almost regular tournament of order n ≥ 7; then e(Tn) = 3. 2. Let k ∈ {n, n + 1, n + 2}. We give the sufficient and necessary conditions for Tn such that e(Tn) = k, and obtain all Tn’s such that e(Tn) = k.

An orientation of a complete graph is a tournament, and an orientation of a complete n-partite graph is an n-partite tournament. For each n 2:: 4, there exist examples of strongly connected n-partite tournament without any strongly connected subtournaments of order p 2:: 4. If D is a digraph, then let d+ (x) be the out degree and d(x) the indegree of the vertex x in D. The minimum (maximum) out...

Let T be a 3-partite tournament. We say that a vertex v is −→ C3 -free if v does not lie on any directed triangle of T . Let F3(T ) be the set of the −→ C3 -free vertices in a 3-partite tournament and f3(T ) its cardinality. In this paper we prove that if T is a regular 3-partite tournament, then F3(T )must be contained in one of the partite sets of T . It is also shown that for every regular 3...

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If x is a vertex of a digraph D, then we denote by d(x) and d−(x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is de6ned by ig(D) = max{d+(x); d−(x)} − min{d+(y); d−(y)} over all vertices x and y o...

Let A be a (0, 1, ∗)-matrix with main diagonal all 0’s and such that if ai,j = 1 or ∗ then aj,i = ∗ or 0. Underwhat conditions on the row sums, and or column sums, of A is it possible to change the ∗’s to 0’s or 1’s and obtain a tournament matrix (the adjacency matrix of a tournament) with a specified score sequence? We answer this question in the case of regular and nearly regular tournaments....