22 research outputs found
Graph parameters from symplectic group invariants
In this paper we introduce, and characterize, a class of graph parameters
obtained from tensor invariants of the symplectic group. These parameters are
similar to partition functions of vertex models, as introduced by de la Harpe
and Jones, [P. de la Harpe, V.F.R. Jones, Graph invariants related to
statistical mechanical models: examples and problems, Journal of Combinatorial
Theory, Series B 57 (1993) 207-227]. Yet they give a completely different class
of graph invariants. We moreover show that certain evaluations of the cycle
partition polynomial, as defined by Martin [P. Martin, Enum\'erations
eul\'eriennes dans les multigraphes et invariants de Tutte-Grothendieck, Diss.
Institut National Polytechnique de Grenoble-INPG; Universit\'e
Joseph-Fourier-Grenoble I, 1977], give examples of graph parameters that can be
obtained this way.Comment: Some corrections have been made on the basis of referee comments. 21
pages, 1 figure. Accepted in JCT
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
On the existence of real R-matrices for virtual link invariants
We characterize the virtual link invariants that can be described as
partition function of a real-valued R-matrix, by being weakly reflection
positive. Weak reflection positivity is defined in terms of joining virtual
link diagrams, which is a specialization of joining virtual link diagram
tangles. Basic techniques are the first fundamental theorem of invariant
theory, the Hanlon-Wales theorem on the decomposition of Brauer algebras, and
the Procesi-Schwarz theorem on inequalities for closed orbits
On partition functions for 3-graphs
A {\em cyclic graph} is a graph with at each vertex a cyclic order of the
edges incident with it specified. We characterize which real-valued functions
on the collection of cubic cyclic graphs are partition functions of a real
vertex model (P. de la Harpe, V.F.R. Jones, Graph invariants related to
statistical mechanical models: examples and problems, Journal of Combinatorial
Theory, Series B 57 (1993) 207--227). They are characterized by `weak
reflection positivity', which amounts to the positive semidefiniteness of
matrices based on the `-join' of cubic cyclic graphs (for all k\in\oZ_+).
Basic tools are the representation theory of the symmetric group and
geometric invariant theory, in particular the Hanlon-Wales theorem on the
decomposition of Brauer algebras and the Procesi-Schwarz theorem on
inequalities defining orbit spaces
Mixed partition functions and exponentially bounded edge-connection rank
We study graph parameters whose associated edge-connection matrices have
exponentially bounded rank growth. Our main result is an explicit construction
of a large class of graph parameters with this property that we call mixed
partition functions. Mixed partition functions can be seen as a generalization
of partition functions of vertex models, as introduced by de la Harpe and
Jones, [P. de la Harpe, V.F.R. Jones, Graph invariants related to statistical
mechanical models: examples and problems, Journal of Combinatorial Theory,
Series B 57 (1993) 207--227] and they are related to invariant theory of
orthosymplectic supergroup. We moreover show that evaluations of the
characteristic polynomial of a simple graph are examples of mixed partition
functions, answering a question of de la Harpe and Jones.Comment: To appear in Ann. Inst. Henri Poincar\'e Comb. Phys. Interac
On the existence of real R-matrices for virtual link invariants
We characterize the virtual link invariants that can be described as partition function of a real-valued R-matrix, by being weakly reflection positive. Weak reflection positivity is defined in terms of joining virtual link diagrams, which is a specialization of joining virtual link diagram tangles. Basic techniques are the first fundamental theorem of invariant theory, the Hanlon–Wales theorem on the decomposition of Brauer algebras, and the Procesi–Schwarz theorem on inequalities for closed orbits