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1.
Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let ik denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x) = ikxk. In particular, we show that if G is a well covered graph with independence number , then all the roots of i(G, x) lie in in the disk |z| (this is far from true if the condition of being well covered is omitted). Moreover, there is a family of well covered graphs (for each ) for which the independence polynomials have a root arbitrarily close to –.  相似文献   

2.
本文讨论了含割点$u$的连通图G,其中$G-u$含路、圈或$D_{n}$分支时图$G$的伴随多项式的最小实根的变化情况.得到一些新的序关系,这推广了文[10-13]中有关图的伴随多项式最小根的一些结果.  相似文献   

3.
The independence polynomial of a (finite) graph is the generating function for the number of independent sets of each cardinality. Assuming that each possible edge of a complete graph of order n is independently operational with probability p, we consider the expected independence polynomial. We show here that for all fixed , the expected independence polynomials of complete graphs have all real, simple roots.  相似文献   

4.
主要讨论了连通图G所含三角形的两个二度点分别与路、圈或Dn(由K3的一个顶点和路的一个端点重迭后所得到的图)相粘接后所得新图的伴随多项式最小根的变化情况,得到一些新结果.  相似文献   

5.
The independence polynomial of a graph G is the function i(G, x) = k0 i k x k, where i k is the number of independent sets of vertices in G of cardinality k. We prove that real roots of independence polynomials are dense in (–, 0], while complex roots are dense in , even when restricting to well covered or comparability graphs. Throughout, we exploit the fact that independence polynomials are essentially closed under graph composition.  相似文献   

6.
Let R be a commutative ring and Г(R) be its zero-divisor graph.We com-pletely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one,two,or three.Furthermore,if R≌ Ri × R2 × … Rn (each Ri is local for i =1,2,3,…,n),we also give algebraic characterizations of the ring R when the clique number of r(R) is four.  相似文献   

7.
Let K be a field. Then there exists a commutative K-algebra A such that each polynomial in K[X] of degree at least 2 has infinitely many roots in A. If B is a finite-dimensional commutative K-algebra and char(K) ≠ 3 (resp., char(K ) = 3), then X 2 + X + 1 (resp., X 2 + X-1) has only finitely many roots in B. Relevant examples are also given, especially of K-algebras of the form K + N, where N is the nilradical.  相似文献   

8.
We study the zeros of two families of polynomials related to rook theory and matchings in graphs. One of these families is based on the cover polynomial of a digraph introduced by Chung and Graham . Another involves a version of the ‘hit polynomial’ of rook theory, but which applies to weighted matchings in (non-bipartite) graphs. For both of these families we prove a result which is analogous to a theorem of the author, K. Ono, and D. G. Wagner, namely that for Ferrers boards the hit polynomial has only real zeros. We also show that for each of these families there is a general conjecture involving arrays of numbers satisfying inequalities which contains these theorems as special cases. We provide evidence for the truth of these conjectures by proving other special cases and discussing computational experiments.  相似文献   

9.
We study the conditions for truncated symmetric products ofmanifolds to be manifolds. In particular, we show that suitablydefined spaces of systems of real roots of real polynomialsare homeomorphic to real projective spaces. 1991 MathematicsSubject Classification 57N99, 26C10.  相似文献   

10.
An independent set in a graph G is a set of pairwise non-adjacent vertices. Let ik(G)denote the number of independent sets of cardinality k in G. Then its generating function ■is called the independence polynomial of G (Gutman and Harary, 1983). In this paper, we introduce a new graph operation called the cycle cover product and formulate its independence polynomial. We also give a criterion for formulating the independence polynomial of a graph. Based on the exact formulas,we prove s...  相似文献   

11.
本文引入了图族伴随多项式的最小根极值,用它刻画了特征标不小于-1的图族伴随多项式的最小根极值,给出了其对应的极图,并由此得到了一些有关这些图族伴随多项式最小根序关系的新结果.  相似文献   

12.
引入伴随多项式是为了从补图的角度研究色多形式,图的伴随多项式的极小根可用于判定色等价图.β(G)表示图G的伴随多项式的极小根.n表示n个顶点的单圈图的集合.分别确定了具有max{β(G)|G∈Ωn}和min{β(G)|G∈Ωn}的所有单圈图.  相似文献   

13.
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  相似文献   

14.
We consider transversal (orthogonal) perturbations of finite-dimensional convex sets and estimate the degree of nonconvexity of resulting sets, i.e. we estimate the nonconvexity of graphs of continuous functions. We prove that a suitable estimate of nonconvexity of graphs over all lines induces a nice estimate of the nonconvexity of graphs of the entire function. Here, the term nice means that in the well-known Michael selection theorem it is possible to replace convex sets of a multivalued mapping by such nonconvex sets. As a corollary, we obtain positive results for polynomials of degree two under some restrictions on coefficients. Our previous results concerned the polynomials of degree one and Lipschitz functions. We show that for a family of polynomials of degree three such estimate of convexity in general does not exist. Moreover, for degree 9 we show that the nonconvexity of the unique polynomial P(x,y)=x9+x3y realizes the worst possible case.  相似文献   

15.
Define the differential operators ?_n for n∈N inductively by ?_1 [f](z)=f(z) and ?_(n+1) [f](z)=f(z)?_n[f](z)+d/dz ?_n[f](z).For a positive integer k≥2 and a positive number δ,let F be the family of functions f meromorphic on domain D■C such that ?_k[f](z)≠0 and |Res(f,a)-j|≥δ for all j∈{0,1,…,k-1} and all simple poles a of f in D.Then F is quasi-normal on D of order 1.  相似文献   

16.
本文引入了图族伴随多项式的最小根极值,用它刻画了特征标不小于$-1$的图族伴随多项式的最小根极值,给出了其对应的极图, 并由此得到了一些有关这些图族伴随多项式最小根序关系的新结果.  相似文献   

17.
Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is basic connected with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ? A as the automorphism of the Grothendieck group K 0(A) induced by the Auslander-Reiten translation τ in the derived category Der(modA) of the module category modA of finite dimensional left A-modules. We say that A is an algebra of cyclotomic type if the characteristic polynomial χ A of ? A is a product of cyclotomic polynomials. There are many examples of algebras of cyclotomic type in the representaton theory literature: hereditary algebras of Dynkin and extended Dynkin types, canonical algebras, some supercanonical and extended canonical algebras. Among other results, we show that: (a) algebras satisfying the fractional Calabi-Yau property have periodic Coxeter transformation and are, therefore, of cyclotomic type, and (b) algebras whose homological form h A is non-negative are of cyclotomic type. For an algebra A of cyclotomic type we describe the shape of the Auslander-Reiten components of Der(modA).  相似文献   

18.
In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of RR is not a radical module for some right coherent rings. We call a ring a right X ring if Homa(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith‘s conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.  相似文献   

19.
In this paper we study the problem of characterizing the real Banach spaces whose unit sphere determines polynomials, i.e., if two polynomials coincide in the unit sphere, is this sufficient to guarantee that they are identical? We show that, in the frame of spaces with unconditional basis, non- reflexivity is a sufficient, although not necessary, condition for the above question to have an affirmative answer. We prove that the only lp^n spaces having this property are those with p irrational, while the only lp spaces which do not enjoy it are those with p an even integer. We also introduce a class of polynomial determining sets in any real Banach space.  相似文献   

20.
This paper starts with an observation that two infinite series of simplicial complexes, which a priori do not seem to have anything to do with each other, have the same homotopy type. One series consists of the complexes of directed forests on a double directed string, while the other one consists of Shapiro–Welker models for the spaces of hyperbolic polynomials with a triple root. We explain this coincidence in the more general context by finding an explicit homotopy equivalence between complexes of directed forests on a double directed tree, and doubly disconnecting complexes of a tree.  相似文献   

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