Designs, matrices, and tournaments
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IAIN CRUMP, Simon Fraser University
An invariant from incidence matrix permanents  [PDF]

We introduce a new invariant for graphs with a particular vertex-edge count, found from the permanent of a matrix built from signed incident matrix blocks. In the special case of 4-regular graphs, we find that this invariant is preserved by the graph theoretic operations that preserve the Feynman period.

RICHARD WILLIAM RAMSAY DARLING, United States Department of Defense
Rank deficiency in sparse random GF[2] matrices  [PDF]

Binary matrix $M$ is random $m \times n$, i.i.d. rows. $N (n,m)$ counts left null vectors in $\{0,1\}^m$ for $M$, mod 2. Take $n, m \to \infty$, with $m/n \to \alpha > 0$; i.i.d. row sums converge to $W \geq 3$. Threshold $\alpha^*$ is infimum of $\alpha$ at which $n^{-1} \log{\mathbb{E} [N (n,m)}]$ has positive limit; $\underline{\alpha} \geq \alpha^*$ is infimum of $\alpha$ so 2-core (endpoint of iterative algorithm that deletes rows incident to singleton columns) has more rows than non-empty columns. Random row weights cause interesting new phenomena.(Darling, Penrose, Wade, Zabell, Electronic J. Probability 19-83, 2014)

ROBERT LUTHER, Memorial University of Newfoundland
Equitably Colourable Combinatorial Designs  [PDF]

Suppose we have a BIBD$(v,k,\lambda)$ whose points are coloured with $\ell$ colours $c_1,...,c_\ell$. A block $B$ is equitably $\ell$-coloured if $B$ has $n_i$ points coloured with colour $c_i$ $(i=1,...,\ell)$ and $|n_i-n_j|\leq 1$ for any $i,j\in \{1,...,\ell\}$. A BIBD is equitably $\ell$-colourable if the points can be coloured with $\ell$ colours such that every block is equitably $\ell$-coloured. The associated spectrum problem is the problem of determining conditions on $v$ such that an equitably $\ell$-colourable BIBD$(v,k,\lambda)$ can and will exist for fixed $\ell$, $k$, and $\lambda$. In this talk we will settle this spectrum problem for any $\ell$, $k$, and $\lambda$.

KYLE WEBB, Wake Forest University
Balance in Generalized Tournaments  [PDF]

In this talk we consider generalizations of standard tournament graphs. Here $n$ players compete in the ${n \choose k}$ possible $k$-player games (where $k\leq n$) and we are interested, for instance, in maximal and minimal numbers of edges for associated graphs, under different edge (domination) constructions. A theorem regarding existence of balanced (zero-edge) generalized tournament graphs is presented.