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1.
The energy of unitary cayley graphs   总被引:1,自引:0,他引:1  
A graph G of order n is called hyperenergetic if E(G)>2n-2, where E(G) denotes the energy of G. The unitary Cayley graph Xn has vertex set Zn={0,1,2,…,n-1} and vertices a and b are adjacent, if gcd(a-b,n)=1. These graphs have integral spectrum and play an important role in modeling quantum spin networks supporting the perfect state transfer. We show that the unitary Cayley graph Xn is hyperenergetic if and only if n has at least two prime factors greater than 2 or at least three distinct prime factors. In addition, we calculate the energy of the complement of unitary Cayley graph and prove that is hyperenergetic if and only if n has at least two distinct prime factors and n≠2p, where p is a prime number. By extending this approach, for every fixed , we construct families of k hyperenergetic non-cospectral integral circulant n-vertex graphs with equal energy.  相似文献   

2.
This paper gives upper and lower bounds of the Christoffel-type functions , for the m-orthogonal polynomials for a Freud weight W=e-Q, which are given as follows. Let an=an(Q) be the nth Mhaskar–Rahmanov–Saff number, φn(x)=max{n-2/3,1-|x|/an}, and d>0. Assume that QC(R) is even, , and for some A,B>1
Then for xR
and for |x|an(1+dn-2/3)
  相似文献   

3.
Let K be a convex body in d (d2), and denote by Bn(K) the set of all polynomials pn in d of total degree n such that |pn|1 on K. In this paper we consider the following question: does there exist a p*nBn(K) which majorates every element of Bn(K) outside of K? In other words can we find a minimal γ1 and p*nBn(K) so that |pn(x)|γ |p*n(x)| for every pnBn(K) and x d\K? We discuss the magnitude of γ and construct the universal majorants p*n for evenn. It is shown that γ can be 1 only on ellipsoids. Moreover, γ=O(1) on polytopes and has at most polynomial growth with respect to n, in general, for every convex body K.  相似文献   

4.
This work is based on ideas of Ili? [A. Ili?, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881-1889] on the energy of unitary Cayley graph. For a finite commutative ring R with unity , the unitary Cayley graph of R is the Cayley graph whose vertex set is R and the edge set is {{a,b}:a,bRanda-bR×}, where R× is the group of units of R. We study the eigenvalues of the unitary Cayley graph of a finite commutative ring and some gcd-graphs and compute their energy. Moreover, we obtain the energy for the complement of unitary Cayley graphs.  相似文献   

5.
The set of all probability measures σ on the unit circle splits into three disjoint subsets depending on properties of the derived set of {|n|2}n0, denoted by Lim(σ). Here {n}n0 are orthogonal polynomials in L2(). The first subset is the set of Rakhmanov measures, i.e., of σ with {m}=Lim(σ), m being the normalized (m( )=1) Lebesgue measure on . The second subset Mar( ) consists of Markoff measures, i.e., of σ with mLim(σ), and is in fact the subject of study for the present paper. A measure σ, belongs to Mar( ) iff there are >0 and l>0 such that sup{|an+j|:0jl}>, n=0,1,2,…,{an} is the Geronimus parameters (=reflectioncoefficients) of σ. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of σ with {m}Lim(σ). We show that σ is ratio asymptotic iff either σ is a Rakhmanov measure or σ satisfies the López condition (which implies σMar( )). Measures σ satisfying Lim(σ)={ν} (i.e., weakly asymptotic measures) are also classified. Either ν is the sum of equal point masses placed at the roots of zn=λ, λ , n=1,2,…, or ν is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism zzn, n=1,2,…, of a closed arc J (including J= ) with removed open concentric arc J0 (including J0=). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures ν and show that these measures satisfy {ν}=Lim(ν).  相似文献   

6.
In the present paper we find a new interpretation of Narayana polynomials Nn(x) which are the generating polynomials for the Narayana numbers where stands for the usual binomial coefficient, i.e. . They count Dyck paths of length n and with exactly k peaks, see e.g. [R.A. Sulanke, The Narayana distribution, in: Lattice Path Combinatorics and Applications (Vienna, 1998), J. Statist. Plann. Inference 101 (1–2) (2002) 311–326 (special issue)] and they appeared recently in a number of different combinatorial situations, see for e.g. [T. Doslic, D. Syrtan, D. Veljan, Enumerative aspects of secondary structures, Discrete Math. 285 (2004) 67–82; A. Sapounakis, I. Tasoulas, P. Tsikouras, Counting strings in Dyck paths, Discrete Math. 307 (2007) 2909–2924; F. Yano, H. Yoshida, Some set partitions statistics in non-crossing partitions and generating functions, Discrete Math. 307 (2007) 3147–3160]. Strangely enough Narayana polynomials also occur as limits as n of the sequences of eigenpolynomials of the Schur–Szeg? composition map sending (n−1)-tuples of polynomials of the form (x+1)n−1(x+a) to their Schur–Szeg? product, see below. We present below a relation between Narayana polynomials and the classical Gegenbauer polynomials which implies, in particular, an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence {Nn(x)}.  相似文献   

7.
Let be a family of polynomials such that , i=1,…,r. We say that the family P has the PSZ property if for any set with there exist infinitely many such that E contains a polynomial progression of the form {a,a+p1(n),…,a+pr(n)}. We prove that a polynomial family P={p1,…,pr} has the PSZ property if and only if the polynomials p1,…,pr are jointly intersective, meaning that for any there exists such that the integers p1(n),…,pr(n) are all divisible by k. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If are jointly intersective integral polynomials, then for any finite partition of , there exist i{1,…,k} and a,nEi such that {a,a+p1(n),…,a+pr(n)}Ei.  相似文献   

8.
9.
If P is a polynomial on Rm of degree at most n, given by , and Pn(Rm) is the space of such polynomials, then we define the polynomial |P| by . Now if is a convex set, we define the norm on Pn(Rm), and then we investigate the inequality providing sharp estimates on for some specific spaces of polynomials. These ’s happen to be the unconditional constants of the canonical bases of the considered spaces.  相似文献   

10.
A (finite or countably infinite) set G of generators of an abstract C*-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space AB(H) and every sequence of unital completely positive linear maps ϕ 1, ϕ 2,... from B(H) to itself,
limn ? ¥ ||fn(g) - g|| = 0,"g ? G T limn ? ¥ fn(a) - a|| = 0,"a ? A.\mathop {\lim }\limits_{n \to \infty } ||{\phi _n}(g) - g|| = 0,{\forall _g} \in G \Rightarrow \mathop {\lim }\limits_{n \to \infty } {\phi _n}(a) - a|| = 0,{\forall _a} \in A.  相似文献   

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