Лагранжевы подпространства, дельта-матроиды и четырехчленные соотношения
We prove that the generating function for the symmetric chromatic polynomial of all simple graphs is (after an appropriate scaling change of variables) a linear combination of one-part Schur polynomials. This statement immediately implies that it is also a tau-function of the Kadomtsev–Petviashvili integrable hierarchy of mathematical physics. Moreover, we describe a large family of polynomial graph invariants leading to the same tau-function. In particular, we introduce the Abel polynomial for graphs and show this for its generating function. The key point here is a Hopf algebra structure on the space spanned by graphs and the behavior of the invariants on its primitive space.
A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassiliev’s theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight system corresponding to the colored Jones polynomial. This weight system can be easily defined in terms of the Lie algebra sl2, but this definition is too cumbersome from the computational point of view, so that the values of this weight system are known only for some limited classes of chord diagrams. In the present paper we give a formula for the values of the sl2 weight system for a class of chord diagrams whose intersection graphs are complete bipartite graphs with no more than three vertices in one of the parts. Our main computational tool is the Chmutov–Varchenko reccurence relation. Furthermore, complete bipartite graphs with no more than three vertices in one of the parts generate Hopf subalgebras of the Hopf algebra of graphs, and we deduce formulas for the projection onto the subspace of primitive elements along the subspace of decomposable elements in these subalgebras. We compute the values of the sl2 weight system for the projections of chord diagrams with such intersection graphs. Our results confirm certain conjectures due to S. K. Lando on the values of the weight system sl2 at the projections of chord diagrams on the space of primitive elements.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.