色多项式对图的刻画

By means of the chromatic polynomials, this paper provided a necessary and sufficient condition for the graph G being a mono-cycle graph(the Theorem 1), a first class bi-cycle graph and a second class bicycle graph(the Theorem 2), respectively.     

Jacobians of Symmetric Polynomials

Annals of Combinatorics - We give the Jacobian of any family of complete symmetric functions, or of power sums, in a finite number of variables.     

Orientations, Lattice Polytopes, and Group Arrangements I: Chromatic and Tension Polynomials of Graphs

This is the first one of a series of papers on association of orientations, lattice polytopes, and group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such a viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the Tutte polynomial T(G; x, y) of a graph G at (1, 0) as the number of cut-equivalence classes of acyclic orientations on G.     

Hermitian Symmetric Polynomials and CR Complexity

Properties of Hermitian forms are used to investigate several natural questions from CR geometry. To each Hermitian symmetric polynomial we assign a Hermitian form. We study how the signature pairs of two Hermitian forms behave under the polynomial product. We show, except for three trivial cases, that every signature pair can be obtained from the product of two indefinite forms. We provide several new applications to the complexity theory of rational mappings between hyperquadrics, including a stability result about the existence of non-trivial rational mappings from a sphere to a hyperquadric with a given signature pair.     

Symmetric Polynomials and Some Good Codes

Elements from extensions of F_q are employed to construct a class of linear codes over F_q with good parameters through symmetric polynomials over F_q.     

Generating Polynomials and Symmetric Tensor Decompositions

This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization of a generating polynomial belongs to the apolar ideal of the tensor. A symmetric tensor decomposition can be determined by a set of generating polynomials, which can be represented by a matrix. We call it a generating matrix. Generally, a symmetric tensor decomposition can be determined by a generating matrix satisfying certain conditions. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose methods for computing symmetric tensor decompositions. Extensive examples are shown to demonstrate the efficiency of proposed methods.     

Symmetric Identities for Euler Polynomials

In this paper we establish two symmetric identities on sums of products of Euler polynomials.     

The L-Appell Symmetric Orthogonal Polynomials

In this paper, we shall be concerned with lowering operators defined on polynomials by means of

$$\begin{aligned} L(x^n)=\mu _nx^{n-1},\ \ n=0,1,\ldots , \ \mu _0=0,\ \ \mu _n\ne 0\ \ (n=1,2,\ldots ). \end{aligned}$$

We determine a necessary and sufficient condition on lowering operators L and a symmetric orthogonal polynomial sets \(\{P_n\}_{n\ge 0}\) such that \(\{P_n\}_{n\ge 0}\) is L-Appell. The resulting polynomials are the generalized Hermite and the symmetric PSs related to Wall and generalized Stieltjes–Wigert. Various properties of the obtained families are singled out: a three-term recurrence relation, explicit expression in term of hypergeometric and basic hypergeometric functions and generating functions.

     

Chromatic Polynomials of Complements of Bipartite Graphs

We define a biclique to be the complement of a bipartite graph, consisting of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic polynomial of an arbitrary biclique, and use this to give certain conditions under which two of the graphs have chromatic polynomials with the same splitting field. Finally, we use a subfamily of bicliques to prove the cubic case of the α + n conjecture, by showing that for any cubic integer α, there is a natural number n such that α + n is a chromatic root.     

Biased Expansions of Biased Graphs and Their Chromatic Polynomials

A biased graph is a graph with a distinguished set of circles, such that if two circles in the set are contained in a theta graph, then so is the third circle of the theta graph. We introduce a new biased graph, a biased expansion of a biased graph, that satisfies certain lifting and projection properties with the original biased graph. We relate the chromatic polynomials of a biased graph and its biased expansions, thus generalizing a biased-graph result of Zaslavsky [7] and a hyperplane result of Ehrenborg and Readdy [1]. We also determine which biased expansions have supersolvable bias matroids.     

Elementary Symmetric Polynomials in Random Variables

The subject of the paper is the probability-theoretic properties of elementary symmetric polynomials σ _k of arbitrary degree k in random variables X _i (i=1,2,…,m) defined on special subsets of commutative rings ℛ _m with identity of finite characteristic m. It is shown that the probability distributions of the random elements σ _k (X ₁,…,X _m ) tend to a limit when m→∞ if X ₁,…,X _m form a Markov chain of finite degree μ over a finite set of states V, V⊂ℛ _m , with positive conditional probabilities. Moreover, if all the conditional probabilities exceed a prescribed positive number α, the limit distributions do not depend on the choice of the chain.      

Symmetric Polynomials on Rearrangement-Invariant Function Spaces

The exact representation of symmetric polynomials on Banachspaces with symmetric basis and also on separable rearrangement-invariantfunction spaces over [0, 1] and [0, ) is given. As a consequenceof this representation it is obtained that, among these spaces,l_2n, L_2n[0, 1], L_2n[0, ) and L_2n[0, )L_2m[0, ) where n, m areboth integers are the only spaces that admit separating polynomials.     

Integer Roots Chromatic Polynomials of Non-Chordal Graphs and the Prouhet-Tarry-Escott Problem

In this paper, we give an affirmative answer to a question of Dmitriev concerning the existence of a non-chordal graph with a chordless cycle of order n whose chromatic polynomial has integer roots for a few values of n, extending prior work of Dong et al. Received: April, 2003     

关于对称拟凸多项式的不相连定理

假定μn为Rn上的标准高斯测度,X为Rn上的随机向量,分布为μn.不相连猜测说的是:如果f与g为Rn上的两个多项式,而且f(X)与g(X)相互独立,则存在Rn上的正交变换Y = LX及整数k使得f o L-1为(y1,y2,…,yk)的函数,goL-1为(yk+1,yk+2,…,yn)的函数.此时,称f与g不相连.在这...     

A Chromatic Symmetric Function in Noncommuting Variables

Stanley (Advances in Math. 111, 1995, 166–194) associated with a graph G a symmetric function X _G which reduces to G's chromatic polynomial under a certain specialization of variables. He then proved various theorems generalizing results about , as well as new ones that cannot be interpreted on the level of the chromatic polynomial. Unfortunately, X _G does not satisfy a Deletion-Contraction Law which makes it difficult to apply the useful technique of induction. We introduce a symmetric function Y _G in noncommuting variables which does have such a law and specializes to X _G when the variables are allowed to commute. This permits us to further generalize some of Stanley's theorems and prove them in a uniform and straightforward manner. Furthermore, we make some progress on the (3 + 1)-free Conjecture of Stanley and Stembridge (J. Combin Theory (A) J. 62, 1993, 261–279).     

On Chromatic Polynomials of Some Kinds of Graphs

In this paper,a new method is used to calculate the chromatic polynomials of graphs.The chro-matic polynomials of the complements of a wheel and a fan are determined.Furthermore,the adjoint polynomialsof F_n with n vertices are obtained.This supports a conjecture put forward by R.Y.Liu et al.     

Polynomials,Symmetric Multilinear Forms and Weak Compactness

In this paper we obtain some versions of weak compactness James theorem, replacing bounded linear functionals by polynomials and symmetric multilinear forms.