4 research outputs found
Finding Fibonacci in the Hyperbolic Plane
No full textWe use a combinatorial approximation of the hyperbolic plane to investigate
properties of hyperbolic geometry such as exponential growth of perimeter and
area of disks, and the linear isoperimetric inequality. This calculations give
a surprising link to Fibonacci numbers.Comment: To appear in Mathematics Magazin
Recommended from our members
Random Groups at Density d<3/14 Act Non-trivially on a CAT(0) Cube Complex
No full textFor random groups in the Gromov density model at d<3/14, we construct walls in the Cayley complex X which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities d<1/5 and d<5/24, respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density d<1/4
Determinants of Seidel Tournament Matrices
No full textThe Seidel matrix of a tournament on players is an
skew-symmetric matrix with entries in that encapsulates the
outcomes of the games in the given tournament. It is known that the determinant
of an Seidel matrix is if is odd, and is an odd perfect
square if is even. This leads to the study of the set \mathcal{D}(n)= \{
\sqrt{\det S}: \mbox{ $S$ is an $n\times n$ Seidel matrix}\}. This paper
studies various questions about . It is shown that
is a proper subset of for every positive
even integer, and every odd integer in the interval is in
for even. The expected value and variance of over
the Seidel matrices chosen uniformly at random is determined, and
upper bounds on are given, and related to the Hadamard
conjecture. Finally, it is shown that for infinitely many ,
contains a gap (that is, there are odd integers such that but ) and several properties of the
characteristic polynomials of Seidel matrices are established.Comment: 21 pages, 7 figure
Graphs with Many Hamiltonian Paths
No full textA graph is \emph{hamiltonian-connected} if every pair of vertices can be
connected by a hamiltonian path, and it is \emph{hamiltonian} if it contains a
hamiltonian cycle. We construct families of non-hamiltonian graphs for which
the ratio of pairs of vertices connected by hamiltonian paths to all pairs of
vertices approaches 1. We then consider minimal graphs that are
hamiltonian-connected. It is known that any order- graph that is
hamiltonian-connected must have edges. We construct an infinite
family of graphs realizing this minimum.Comment: v3: substantial re-writing, including new author. To appear in
Involve. v2: 12 pages, 6 figures. Substantial re-write including new results
and removing results already proven by others. v1: 16 pages, 7 figure
