435 research outputs found
Half-regular factorizations of the complete bipartite graph
We consider a bipartite version of the color degree matrix problem. A bipartite graph G(U,V,E) is half-regular if all vertices in U have the same degree. We give necessary and sufficient conditions for a bipartite degree matrix (also known as demand matrix) to be the color degree matrix of an edge-disjoint union of half-regular graphs. We also give necessary and sufficient perturbations to transform realizations of a half-regular degree matrix into each other. Based on these perturbations, a Markov chain Monte Carlo method is designed in which the inverse of the acceptance ratios is polynomial bounded. Realizations of a half-regular degree matrix are generalizations of Latin squares, and they also appear in applied neuroscience. © 201
The Zeta Function of a Hypergraph
We generalize the Ihara-Selberg zeta function to hypergraphs in a natural
way. Hashimoto's factorization results for biregular bipartite graphs apply,
leading to exact factorizations. For -regular hypergraphs, we show that
a modified Riemann hypothesis is true if and only if the hypergraph is
Ramanujan in the sense of Winnie Li and Patrick Sol\'e. Finally, we give an
example to show how the generalized zeta function can be applied to graphs to
distinguish non-isomorphic graphs with the same Ihara-Selberg zeta function.Comment: 24 pages, 6 figure
Hurwitz equivalence of braid monodromies and extremal elliptic surfaces
We discuss the equivalence between the categories of certain ribbon graphs
and subgroups of the modular group and use it to construct
exponentially large families of not Hurwitz equivalent simple braid monodromy
factorizations of the same element. As an application, we also obtain
exponentially large families of {\it topologically} distinct algebraic objects
such as extremal elliptic surfaces, real trigonal curves, and real elliptic
surfaces
A simple model of trees for unicellular maps
We consider unicellular maps, or polygon gluings, of fixed genus. A few years
ago the first author gave a recursive bijection transforming unicellular maps
into trees, explaining the presence of Catalan numbers in counting formulas for
these objects. In this paper, we give another bijection that explicitly
describes the "recursive part" of the first bijection. As a result we obtain a
very simple description of unicellular maps as pairs made by a plane tree and a
permutation-like structure. All the previously known formulas follow as an
immediate corollary or easy exercise, thus giving a bijective proof for each of
them, in a unified way. For some of these formulas, this is the first bijective
proof, e.g. the Harer-Zagier recurrence formula, the Lehman-Walsh formula and
the Goupil-Schaeffer formula. We also discuss several applications of our
construction: we obtain a new proof of an identity related to covered maps due
to Bernardi and the first author, and thanks to previous work of the second
author, we give a new expression for Stanley character polynomials, which
evaluate irreducible characters of the symmetric group. Finally, we show that
our techniques apply partially to unicellular 3-constellations and to related
objects that we call quasi-constellations.Comment: v5: minor revision after reviewers comments, 33 pages, added a
refinement by degree of the Harer-Zagier formula and more details in some
proof
- …