The n × n Brualdi-Li matrix Bn has recently been shown to have maximal Perron value (spectral radius) ρ among all tournament matrices of even order n, thus settling the conjecture by the same name. This renews our interest in estimating ρ and motivates us to study the Perron eigenvector x of Bn, which is normalized to have 1-norm equal to one. It follows that x minimizes the 2-norm among all Pe...

Knockout tournaments provide a common and important framework for structuring sporting competitions, worth billions of dollars every year in the global economy. Knockout tournaments also model a specific type of election scheme: namely, sequential pairwise elimination elections. The designer of a tournament typically controls both the shape of the tournament (usually a binary tree) and the seed...

Given a tournament T = (V, A), with each subset X of V is associated the subtournament T [X] = (X, A∩(X×X)) of T induced by X. A subset I of V is an interval of T provided that for any a, b ∈ I and x ∈ V \I, (a, x) ∈ A if and only if (b, x) ∈ A. For example, ∅, {x}, where x ∈ V , and V are intervals of T called trivial. A tournament is indecomposable if all its intervals are trivial; otherwise,...

A tournament Tn is an orientation of a complete graph on n vertices. A king in a tournament is a vertex from which every other vertex is reachable by a path of length at most 2. A sorted sequence of kings in a tournament Tn is an ordered list of its vertices u1, u2, . . . , un such that ui dominates ui+1 (ui → ui+1) and ui is a king in the subtournament induced by {uj : i ≤ j ≤ n} for each i = ...

Abstract: A hypertournament or a k-tournament, on n vertices, 2≤k≤n, is a pair T= (V,E), where the vertex setV is a set of size n and the edge setE is the collection of all possible subsets of size k of V, called the edges, each taken in one of its k! possible permutations. A k-tournament is pancyclic if there exists (directed) cycles of all possible lengths; it is vertex-pancyclic if moreover ...

Recently, an electronic sealed-bid auction using cryptography techniques has been proposed. An electronic sealed-bid auction is a type of electronic auctions. The aim of an auction including an electronic sealed-bid auction is to decide the price of goods fairly. So, the decision method of the price on goods in an electronic auction is very important scheme. In this paper, we propose the electr...

The feature selection characterized by relatively small sample size and extremely high dimensional feature space is common in many areas of contemporary statistics. The high dimensionality of the feature space causes serious difficulties: (i) the sample correlations between features become high even if the features are stochastically independent; (ii) the computation becomes intractable. These ...

We study some problems pertaining to the tournament equilibrium set (τ for short). A tournament H is a τ-retentive tournament if there is a tournament T which has a minimal τ-retentive set R such that T [R] is isomorphic to H. We study τ-retentive tournaments and achieve many significant results. In particular, we prove that there are no τretentive tournaments of size 4, only 2 non-isomorphic τ...