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