Loopy, Hankel, and Combinatorially Skew-Hankel Tournaments

We investigate tournaments with a specified score vector having additional structure: loopy tournaments in which loops are allowed, Hankel tournaments which are tournaments symmetric about the Hankel diagonal (the anti-diagonal), and combinatorially skew-Hankel tournaments which are skew-symmetric about the Hankel diagonal. In each case, we obtain necessary and sufficient conditions for existence, algorithms for construction, and switches which allow one to move from any tournament of its type to any other, always staying within the defined type

Reconstruction of complete interval tournaments

Let $a, b$ and $n$ be nonnegative integers $(b \geq a, \ b > 0, \ n \geq 1)$, $\mathcal{G}_n(a,b)$ be a multigraph on $n$ vertices in which any pair of vertices is connected with at least $a$ and at most $b$ edges and \textbf{v =} $(v_1, v_2, ..., v_n)$ be a vector containing $n$ nonnegative integers. We give a necessary and sufficient condition for the existence of such orientation of the edges of $\mathcal{G}_n(a,b)$, that the resulted out-degree vector equals to \textbf{v}. We describe a reconstruction algorithm. In worst case checking of \textbf{v} requires $\Theta(n)$ time and the reconstruction algorithm works in $O(bn^3)$ time. Theorems of H. G. Landau (1953) and J. W. Moon (1963) on the score sequences of tournaments are special cases $b = a = 1$ resp. $b = a \geq 1$ of our result

On realization graphs of degree sequences

Given the degree sequence $d$ of a graph, the realization graph of $d$ is the graph having as its vertices the labeled realizations of $d$, with two vertices adjacent if one realization may be obtained from the other via an edge-switching operation. We describe a connection between Cartesian products in realization graphs and the canonical decomposition of degree sequences described by R.I. Tyshkevich and others. As applications, we characterize the degree sequences whose realization graphs are triangle-free graphs or hypercubes.Comment: 10 pages, 5 figure

Reducing the Effects of Unequal Number of Games on Rankings

Ranking is an important mathematical process in a variety of contexts such as information retrieval, sports and business. Sports ranking methods can be applied both in and beyond the context of athletics. In both settings, once the concept of a game has been defined, teams (or individuals) accumulate wins, losses, and ties, which are then factored into the ranking computation. Many settings involve an unequal number of games between competitors. This paper demonstrates how to adapt two sports rankings methods, the Colley and Massey ranking methods, to settings where an unequal number of games are played between the teams. In such settings, the standard derivations of the methods can produce nonsensical rankings. This paper introduces the idea of including a super-user into the rankings and considers the effect of this fictitious player on the ratings. We apply such techniques to rank batters and pitchers in Major League baseball, professional tennis players, and participants in a free online social game. The ideas introduced in this paper can further the scope that such methods are applied and the depth of insight they offer

Integral matrices with given row and column sums

AbstractLet P = (pij) and Q = (qij) be m × n integral matrices, R and S be integral vectors. Let UPQ(R, S) denote the class of all m × n integral matrices A with row sum vector R and column sum vector S satisfying P ⩽ A ⩽ Q. For a wide variety of classes UPQ(R, S) satisfying our main condition, we obtain two necessary and sufficient conditions for the existence of a matrix in UPQ(R, S). The first characterization unifies the results of Gale-Ryser, Fulkerson, and Anstee. Many other properties of (0, 1)-matrices with prescribed row and column sum vectors generalize to integral classes satisfying the main condition. We also study the decomposibility of integral classes satisfying the main condition. As a consequence of our decomposibility theorem, it follows a theorem of Beineke and Harary on the existence of a strongly connected digraph with given indegree and outdegree sequences. Finally, we introduce the incidence graph for a matrix in an integral class UPQ(R, S) and study the invariance of an element in a matrix in terms of its incidence graph. Analogous to Minty's Lemma for arc colorings of a digraph, we give a very simple labeling algorithm to determine if an element in a matrix is invariant. By observing the relationship between invariant positions of a matrix and the strong connectedness of its incidence graph, we present a very short graph theoretic proof of a theorem of Brualdi and Ross on invariant sets of (0, 1)-matrices. Our proof also implies an analogous theorem for a class of tournament matrices with given row sum vector, as conjectured by the analogy between bipartite tournaments and ordinary tournaments

Proposed Statistical model for Scoring and Ranking Sport Tournaments

A class of modification is proposed for calculating a score for each Player/team in Unbalanced Incomplete paired Comparisons Sports Tournaments. Many papers dealing with Balanced Incomplete Paired Comparison Sports Tournaments with at most one comparison per pair have appeared since 1950. However, little has been written about unbalanced situations in which the player /the team (object) ( j ) plays unequal number of games against the player/the team( m ) in a tournament, and the results of all games can be summarized in a Win-Lose matrix Y = { Yjm } , where Yjm = 1,0,1/2, respectively, according to as the player or the team ( j ) wins, losses or draws against the player or the team (m ). Published papers by Ramanujacharyulu (1964), Cowden, D.J. (1975), and David, H. A.(1988) have concentrated on the problem of converting the results of unbalanced incomplete paired comparison tournaments into rank with little consideration of the main relative ability on each player or team. We suggest (modification) another way of quantifying the outcomes of the games/tournaments, in particular, ratings on a scales, 0 to 5, 1 to 10 .ect. It is important to consider not only the vector Vj(d) or the vectors Sj, in scoring and ranking the k teams in such tournaments, but also the vector Zj, where Zj = Sj + SjRj, to take into account the ratio of the relative ability of each team ( Rj ). The proposed modification helps to introduce these methods for use in comparisons/games (tournaments), where the player/team are quantified on a special scale. e.g. 0-5, 1-10, ..etc. We conclude the following:- The scores stabilized to three decimal places at iteration 2 in Cowden’s method Vj(d) .see table(1.4). The scores stabilized to three decimal places at iteration 2 in David’s method Sj , and it’s modification Zj. The proposed modification (Zj) has the advantage of removing ties from David’s method (Sj), and hence it is the best method