1,954 research outputs found
Bounds on the Coefficients of Tension and Flow Polynomials
The goal of this article is to obtain bounds on the coefficients of modular
and integral flow and tension polynomials of graphs. To this end we make use of
the fact that these polynomials can be realized as Ehrhart polynomials of
inside-out polytopes. Inside-out polytopes come with an associated relative
polytopal complex and, for a wide class of inside-out polytopes, we show that
this complex has a convex ear decomposition. This leads to the desired bounds
on the coefficients of these polynomials.Comment: 16 page
Spherical sets avoiding a prescribed set of angles
Let be any subset of the interval . A subset of the unit
sphere in will be called \emph{-avoiding} if for any
. The problem of determining the maximum surface measure of a -avoiding set was first stated in a 1974 note by Witsenhausen; there the
upper bound of times the surface measure of the sphere is derived from a
simple averaging argument. A consequence of the Frankl-Wilson theorem is that
this fraction decreases exponentially, but until now the upper bound for
the case has not moved. We improve this bound to using an
approach inspired by Delsarte's linear programming bounds for codes, combined
with some combinatorial reasoning. In the second part of the paper, we use
harmonic analysis to show that for there always exists an
-avoiding set of maximum measure. We also show with an example that a
maximiser need not exist when .Comment: 21 pages, 3 figure
Lower Bounds and Series for the Ground State Entropy of the Potts Antiferromagnet on Archimedean Lattices and their Duals
We prove a general rigorous lower bound for
, the exponent of the ground state
entropy of the -state Potts antiferromagnet, on an arbitrary Archimedean
lattice . We calculate large- series expansions for the exact
and compare these with our lower bounds on
this function on the various Archimedean lattices. It is shown that the lower
bounds coincide with a number of terms in the large- expansions and hence
serve not just as bounds but also as very good approximations to the respective
exact functions for large on the various lattices
. Plots of are given, and the general dependence on
lattice coordination number is noted. Lower bounds and series are also
presented for the duals of Archimedean lattices. As part of the study, the
chromatic number is determined for all Archimedean lattices and their duals.
Finally, we report calculations of chromatic zeros for several lattices; these
provide further support for our earlier conjecture that a sufficient condition
for to be analytic at is that is a regular
lattice.Comment: 39 pages, Revtex, 9 encapsulated postscript figures, to appear in
Phys. Rev.
On the chromatic roots of generalized theta graphs
The generalized theta graph \Theta_{s_1,...,s_k} consists of a pair of
endvertices joined by k internally disjoint paths of lengths s_1,...,s_k \ge 1.
We prove that the roots of the chromatic polynomial $pi(\Theta_{s_1,...,s_k},z)
of a k-ary generalized theta graph all lie in the disc |z-1| \le [1 + o(1)]
k/\log k, uniformly in the path lengths s_i. Moreover, we prove that
\Theta_{2,...,2} \simeq K_{2,k} indeed has a chromatic root of modulus [1 +
o(1)] k/\log k. Finally, for k \le 8 we prove that the generalized theta graph
with a chromatic root that maximizes |z-1| is the one with all path lengths
equal to 2; we conjecture that this holds for all k.Comment: LaTex2e, 25 pages including 2 figure
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