31,637 research outputs found
Weighted Well-Covered Claw-Free Graphs
A graph G is well-covered if all its maximal independent sets are of the same
cardinality. Assume that a weight function w is defined on its vertices. Then G
is w-well-covered if all maximal independent sets are of the same weight. For
every graph G, the set of weight functions w such that G is w-well-covered is a
vector space. Given an input claw-free graph G, we present an O(n^6)algortihm,
whose input is a claw-free graph G, and output is the vector space of weight
functions w, for which G is w-well-covered. A graph G is equimatchable if all
its maximal matchings are of the same cardinality. Assume that a weight
function w is defined on the edges of G. Then G is w-equimatchable if all its
maximal matchings are of the same weight. For every graph G, the set of weight
functions w such that G is w-equimatchable is a vector space. We present an
O(m*n^4 + n^5*log(n)) algorithm which receives an input graph G, and outputs
the vector space of weight functions w such that G is w-equimatchable.Comment: 14 pages, 1 figur
Tropical mirror symmetry for elliptic curves
Mirror symmetry relates Gromov-Witten invariants of an elliptic curve with
certain integrals over Feynman graphs. We prove a tropical generalization of
mirror symmetry for elliptic curves, i.e., a statement relating certain labeled
Gromov-Witten invariants of a tropical elliptic curve to more refined Feynman
integrals. This result easily implies the tropical analogue of the mirror
symmetry statement mentioned above and, using the necessary Correspondence
Theorem, also the mirror symmetry statement itself. In this way, our tropical
generalization leads to an alternative proof of mirror symmetry for elliptic
curves. We believe that our approach via tropical mirror symmetry naturally
carries the potential of being generalized to more adventurous situations of
mirror symmetry. Moreover, our tropical approach has the advantage that all
involved invariants are easy to compute. Furthermore, we can use the techniques
for computing Feynman integrals to prove that they are quasimodular forms.
Also, as a side product, we can give a combinatorial characterization of
Feynman graphs for which the corresponding integrals are zero. More generally,
the tropical mirror symmetry theorem gives a natural interpretation of the
A-model side (i.e., the generating function of Gromov-Witten invariants) in
terms of a sum over Feynman graphs. Hence our quasimodularity result becomes
meaningful on the A-model side as well. Our theoretical results are
complemented by a Singular package including several procedures that can be
used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman
graphs.Comment: comment on historical development adde
- …