48 research outputs found
Rainbow eulerian multidigraphs and the product of cycles
An arc colored eulerian multidigraph with colors is rainbow eulerian if
there is an eulerian circuit in which a sequence of colors repeats. The
digraph product that refers the title was introduced by Figueroa-Centeno et al.
as follows: let be a digraph and let be a family of digraphs such
that for every . Consider any function
. Then the product is the
digraph with vertex set and if and only if and .
In this paper we use rainbow eulerian multidigraphs and permutations as a way
to characterize the -product of oriented cycles. We study the
behavior of the -product when applied to digraphs with unicyclic
components. The results obtained allow us to get edge-magic labelings of graphs
formed by the union of unicyclic components and with different magic sums.Comment: 12 pages, 5 figure
Grassmann Integral Representation for Spanning Hyperforests
Given a hypergraph G, we introduce a Grassmann algebra over the vertex set,
and show that a class of Grassmann integrals permits an expansion in terms of
spanning hyperforests. Special cases provide the generating functions for
rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All
these results are generalizations of Kirchhoff's matrix-tree theorem.
Furthermore, we show that the class of integrals describing unrooted spanning
(hyper)forests is induced by a theory with an underlying OSP(1|2)
supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J.
Phys.
On distinguishing trees by their chromatic symmetric functions
Let be an unrooted tree. The \emph{chromatic symmetric function} ,
introduced by Stanley, is a sum of monomial symmetric functions corresponding
to proper colorings of . The \emph{subtree polynomial} , first
considered under a different name by Chaudhary and Gordon, is the bivariate
generating function for subtrees of by their numbers of edges and leaves.
We prove that , where is the Hall inner
product on symmetric functions and is a certain symmetric function that
does not depend on . Thus the chromatic symmetric function is a stronger
isomorphism invariant than the subtree polynomial. As a corollary, the path and
degree sequences of a tree can be obtained from its chromatic symmetric
function. As another application, we exhibit two infinite families of trees
(\emph{spiders} and some \emph{caterpillars}), and one family of unicyclic
graphs (\emph{squids}) whose members are determined completely by their
chromatic symmetric functions.Comment: 16 pages, 3 figures. Added references [2], [13], and [15
The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions
We consider a large class of self-adjoint elliptic problem associated with
the second derivative acting on a space of vector-valued functions. We present
two different approaches to the study of the associated eigenvalues problems.
The first, more general one allows to replace a secular equation (which is
well-known in some special cases) by an abstract rank condition. The latter
seems to apply particularly well to a specific boundary condition, sometimes
dubbed "anti-Kirchhoff" in the literature, that arise in the theory of
differential operators on graphs; it also permits to discuss interesting and
more direct connections between the spectrum of the differential operator and
some graph theoretical quantities. In either case our results yield, among
other, some results on the symmetry of the spectrum
On The Growth Of Permutation Classes
We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order.
First, we consider monotone grid classes of permutations. We present procedures for calculating the generating function of any class whose matrix has dimensions m × 1 for some m, and of acyclic and unicyclic classes of gridded permutations. We show that almost all large permutations in a grid class have the same shape, and determine this limit shape.
We prove that the growth rate of a grid class is given by the square of the spectral radius of an associated graph and deduce some facts relating to the set of grid class growth rates. In the process, we establish a new result concerning tours on graphs. We also prove a similar result relating the growth rate of a geometric grid class to the matching polynomial of a graph, and determine the effect of edge subdivision on the matching polynomial. We characterise the growth rates of geometric grid classes in terms of the spectral radii of trees.
We then investigate the set of growth rates of permutation classes and establish a new upper bound on the value above which every real number is the growth rate of some permutation class. In the process, we prove new results concerning expansions of real numbers in non-integer bases in which the digits are drawn from sets of allowed values.
Finally, we introduce a new enumeration technique, based on associating a graph with each permutation, and determine the generating functions for some previously unenumerated classes. We conclude by using this approach to provide an improved lower bound on the growth rate of the class of permutations avoiding the pattern 1324. In the process, we prove that, asymptotically, patterns in Łukasiewicz paths exhibit a concentrated Gaussian distribution
On distinguishing trees by their chromatic symmetric functions
This is the author's accepted manuscript