348 research outputs found
Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits
We deploy algebraic complexity theoretic techniques for constructing
symmetric determinantal representations of for00504925mulas and weakly skew
circuits. Our representations produce matrices of much smaller dimensions than
those given in the convex geometry literature when applied to polynomials
having a concise representation (as a sum of monomials, or more generally as an
arithmetic formula or a weakly skew circuit). These representations are valid
in any field of characteristic different from 2. In characteristic 2 we are led
to an almost complete solution to a question of B\"urgisser on the
VNP-completeness of the partial permanent. In particular, we show that the
partial permanent cannot be VNP-complete in a finite field of characteristic 2
unless the polynomial hierarchy collapses.Comment: To appear in the AMS Contemporary Mathematics volume on
Randomization, Relaxation, and Complexity in Polynomial Equation Solving,
edited by Gurvits, Pebay, Rojas and Thompso
Homomorphisms, representations and characteristic polynomials of digraphs
AbstractThe existence of a homomorphism between two digraphs often implies many structural properties. We collect in this paper some characterizations of various digraph homomorphisms using matrix equations and fiber partitions. We also survey the relationship among the characteristic polynomials of a digraph and its divisors. This includes an introduction of the concept of branched coverings of digraphs, their voltage assignment representations, and a decomposition formula for the characteristic polynomial of a branched cover with branch index 1. Some open problems are included
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs
AbstractFor a nonnegative n×n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A)=n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n×n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials
Decomposition formulas of zeta functions of graphs and digraphs
AbstractWe give a decomposition formula of the zeta function of a regular covering of a graph G with respect to equivalence classes of prime, reduced cycles of G. Furthermore, we give a decomposition formula of the zeta function of a g-cyclic Γ-cover of a symmetric digraph D with respect to equivalence classes of prime cycles of D, for any finite group Γ and g∈Γ
Small covers and the equivariant bordism classification of 2-torus manifolds
Associated with the Davis-Januszkiewicz theory of small covers, this paper
deals with the theory of 2-torus manifolds from the viewpoint of equivariant
bordism. We define a differential operator on the "dual" algebra of the
unoriented -representation algebra introduced by Conner and Floyd, where
. With the help of -colored graphs (or mod 2 GKM graphs), we
may use this differential operator to give a very simple description of tom
Dieck-Kosniowski-Stong localization theorem in the setting of 2-torus
manifolds. We then apply this to study the -equivariant unoriented bordism
classification of -dimensional 2-torus manifolds. We show that the
-equivariant unoriented bordism class of each -dimensional 2-torus
manifold contains an -dimensional small cover as its representative, solving
the conjecture posed in [19]. In addition, we also obtain that the graded
noncommutative ring formed by the equivariant unoriented bordism classes of
2-torus manifolds of all possible dimensions is generated by the classes of all
generalized real Bott manifolds (as special small covers over the products of
simplices). This gives a strong connection between the computation of
-equivariant bordism groups or ring and the Davis-Januszkiewicz theory of
small covers. As a computational application, with the help of computer, we
completely determine the structure of the group formed by equivariant bordism
classes of all 4-dimensional 2-torus manifolds. Finally, we give some essential
relationships among 2-torus manifolds, coloring polynomials, colored simple
convex polytopes, colored graphs.Comment: 32 pages, updated version with the title of paper changed and a large
expansio
Chromatic Polynomials and Rings in Species
Abstract. We present a generalization of the chromatic polynomial, and chromatic symmetric function, arising in the study of combinatorial species. These invariants are defined for modules over lattice rings in species. The primary examples are graphs and set partitions. For these new invariants, we present analogues of results regarding stable partitions, the bond lattice, the deletion-contraction recurrence, and the subset expansion formula. We also present two detailed examples, one related to enumerating subgraphs by their blocks, and a second example related to enumerating subgraphs of a directed graph by their strongly connected components. Resumé. Nous présentons une généralisation du polynôme chromatique et de la fonction symétrique chromatique, qui apparaissent dans l’étude des espèces de structures. Ces invariants sont définis pour modules sur anneaux réticulés aux espéces de structures. Les exemples principaux sont les graphes et les partitions d’entiers. Pour ces invariants nouveaux, nous présentons d’analogues de rsultats concernants les partitions stables, le treillis de liaisons, la rélation de contraction-suppression, et la formule d’expansion en termes de sous-ensembles. Nous présentons aussi deux exemples détaill´s, l’un lié à l’énumération des sous-graphes par ses blocs, et l’autre lié à l’énumération des sousgraphes d’un graphe dirigé par ses composantes fortement connexes
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