Local Out-Tournaments with Upset Tournament Strong Components I: Full and Equal {0,1}-Matrix Ranks

A digraph D is a local out-tournament if the outset of every vertex is a tournament. Here, we use local out-tournaments, whose strong components are upset tournaments, to explore the corresponding ranks of the adjacency matrices. Of specific interest is the out-tournament whose adjacency matrix has boolean, nonnegative integer, term, and real rank all equal to the number of vertices, n. Corresponding results for biclique covers and partitions of the digraph are provided

The conjugacy problem for automorphism groups of countable homogeneous structures

We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility

On the number of 4-cycles in a tournament

If $T$ is an $n$-vertex tournament with a given number of $3$-cycles, what can be said about the number of its $4$-cycles? The most interesting range of this problem is where $T$ is assumed to have $c\cdot n^3$ cyclic triples for some $c>0$ and we seek to minimize the number of $4$-cycles. We conjecture that the (asymptotic) minimizing $T$ is a random blow-up of a constant-sized transitive tournament. Using the method of flag algebras, we derive a lower bound that almost matches the conjectured value. We are able to answer the easier problem of maximizing the number of $4$-cycles. These questions can be equivalently stated in terms of transitive subtournaments. Namely, given the number of transitive triples in $T$, how many transitive quadruples can it have? As far as we know, this is the first study of inducibility in tournaments.Comment: 11 pages, 5 figure

Small World Network of Athletes: Graph Representation of the World Professional Tennis Player

The paper proposes an alternative way to observe and extract the multiple matches games of sports, i.e.: tennis tournament in the Athlete’s Historical Relative Performance Index and its representation as graph. The finding of the small world topology is elaborated along with further statistical patterns in the fashion of the weighted and directed network. The explanation of the sport tournament system as a highly optimized system is hypothetically proposed. Finally, some elaborations regarding to further directions of the usability of the proposed methodology is discussed