173 research outputs found
The WDVV equations in pure Seiberg-Witten theory
We review the relationship between pure four-dimensional Seiberg–Witten theory and the periodic Toda chain. We discuss the definition of the prepotential and give two proofs that it satisfies the generalized Witten–Dijkgraaf–Verlinde–Verlinde equations. A number of steps in the definitions and proofs that is missing in the literature is supplied
Connectivity of generating graphs of nilpotent groups
Let be -generated group. The generating graph of is the
graph whose vertices are the elements of and where two vertices and
are adjacent if . This graph encodes the combinatorial
structure of the distribution of generating pairs across . In this paper we
study several natural graph theoretic properties related to the connectedness
of in the case where is a finite nilpotent group. For example,
we prove that if is nilpotent, then the graph obtained from by
removing its isolated vertices is maximally connected and, if ,
also Hamiltonian. We pose several questions.Comment: 11 pages; to appear in Algebraic Combinatoric
A Survey on Monochromatic Connections of Graphs
The concept of monochromatic connection of graphs was introduced by Caro and
Yuster in 2011. Recently, a lot of results have been published about it. In
this survey, we attempt to bring together all the results that dealt with it.
We begin with an introduction, and then classify the results into the following
categories: monochromatic connection coloring of edge-version, monochromatic
connection coloring of vertex-version, monochromatic index, monochromatic
connection coloring of total-version.Comment: 26 pages, 3 figure
Cut-and-join structure and integrability for spin Hurwitz numbers
Spin Hurwitz numbers are related to characters of the Sergeev group, which
are the expansion coefficients of the Q Schur functions, depending on odd times
and on a subset of all Young diagrams. These characters involve two dual
subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur
functions Q_R with R\in SP are common eigenfunctions of cut-and-join operators
W_\Delta with \Delta\in OP. The eigenvalues of these operators are the
generalized Sergeev characters, their algebra is isomorphic to the algebra of Q
Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the
generating function of spin Hurwitz numbers is a \tau-function of an integrable
hierarchy, that is, of the BKP type. At last, we discuss relations of the
Sergeev characters with matrix models.Comment: 22 page
- …