Topology
and Geometry Seminar
Speaker
: Dr. Jun Ge, Sichuan Normal
University
Time: 3:30-5:30pm Wednesday,
Oct 19, 2016
Place: X2511, Xipu campus
SWJTU
Title: Kenyon's conjecture
Abstract: In graph theory, $\tau(G)$, the number of
spanning trees of a graph $G$ (with $v(G)$ vertices and $e(G)$ edges), usually
plays a central role. The tree entropy, $\frac{\log \tau(G)}{v(G)}$ has been
extensively studied, especially for lattices. We call $\frac{\log
\tau(G)}{e(G)}$ the edge spanning tree entropy of graph $G$. In 1996, Kenyon
raised a conjecture on the upper bound of the edge-spanning tree density of a
planar graph when studying tiling a rectangle with fewest squares.
Interestingly, it can be translated into the language of knot theory, which is
called the determinant density conjecture for knots and links. Both the graph
theory version and the knot theory version of the conjecture are now named
after Kenyon. In this talk, we will first sketch the combinatorial and
geometrical backgrounds of Kenyon's conjecture. Then we will point out that a
Stoimenow's result can give the best known upper bound for edge spanning tree
entropy of planar graphs and determinant density of links. We will verify the
Konyon's conjecture for links with at most 47 crossings, and equivalently, for
planar graphs with at most 47 edges. Special classes such as torus links,
Montesinos links, generalized theta graphs,
outerplanar graphs, 2D Archimedean lattices and Lave
lattices, 3-regular graphs will also be verified. The forbidden structures in
minimal counter-examples will also be mentioned.
