218 research outputs found
Characterizing partition functions of the edge-coloring model by rank growth
We characterize which graph invariants are partition functions of an
edge-coloring model over the complex numbers, in terms of the rank growth of
associated `connection matrices'
On traces of tensor representations of diagrams
Let be a set, of {\em types}, and let \iota,o:T\to\oZ_+. A {\em
-diagram} is a locally ordered directed graph equipped with a function
such that each vertex of has indegree
and outdegree . (A directed graph is {\em locally ordered} if at
each vertex , linear orders of the edges entering and of the edges
leaving are specified.)
Let be a finite-dimensional \oF-linear space, where \oF is an
algebraically closed field of characteristic 0. A function on assigning
to each a tensor is called a {\em tensor representation} of . The {\em trace} (or {\em
partition function}) of is the \oF-valued function on the
collection of -diagrams obtained by `decorating' each vertex of a
-diagram with the tensor , and contracting tensors along
each edge of , while respecting the order of the edges entering and
leaving . In this way we obtain a {\em tensor network}.
We characterize which functions on -diagrams are traces, and show that
each trace comes from a unique `strongly nondegenerate' tensor representation.
The theorem applies to virtual knot diagrams, chord diagrams, and group
representations
Finding k partially disjoint paths in a directed planar graph
The {\it partially disjoint paths problem} is: {\it given:} a directed graph,
vertices , and a set of pairs from
, {\it find:} for each a directed path
such that if then and are disjoint.
We show that for fixed , this problem is solvable in polynomial time if
the directed graph is planar. More generally, the problem is solvable in
polynomial time for directed graphs embedded on a fixed compact surface.
Moreover, one may specify for each edge a subset of
prescribing which of the paths are allowed to traverse this edge
The Strong Arnold Property for 4-connected flat graphs
We show that if is a 4-connected flat graph, then any real
symmetric matrix with exactly one negative eigenvalue and
satisfying, for any two distinct vertices and , if and
are adjacent, and if and are nonadjacent, has the Strong
Arnold Property: there is no nonzero real symmetric matrix with
and whenever and are equal or adjacent. (A graph
is {\em flat} if it can be embedded injectively in -dimensional Euclidean
space such that the image of any circuit is the boundary of some disk disjoint
from the image of the remainder of the graph.)
This applies to the Colin de Verdi\`ere graph parameter, and extends similar
results for 2-connected outerplanar graphs and 3-connected planar graphs
Semidefinite bounds for nonbinary codes based on quadruples
For nonnegative integers , let denote the maximum
cardinality of a code of length over an alphabet with letters and
with minimum distance at least . We consider the following upper bound on
. For any , let \CC_k be the collection of codes of cardinality
at most . Then is at most the maximum value of
, where is a function \CC_4\to R_+ such that
and if has minimum distance less than , and
such that the \CC_2\times\CC_2 matrix (x(C\cup C'))_{C,C'\in\CC_2} is
positive semidefinite. By the symmetry of the problem, we can apply
representation theory to reduce the problem to a semidefinite programming
problem with order bounded by a polynomial in . It yields the new upper
bounds , , , and
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