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Denote by D(G)=(di,j)n×n the distance matrix of a connected graph G with n vertices, where dij is equal to the distance between vertices vi and vj in G . The least eigenvalue of D(G) is called the least distance eigenvalue of G , denoted by λn. In this paper, we determine all the graphs with λn∈[−2.383,0]. 相似文献
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A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G-v is less than the total domination number of G . We call these graphs γt-critical. If such a graph G has total domination number k, we call it k -γt-critical. We verify an open problem of k -γt-critical graphs and obtain some results on the characterization of total domination critical graphs of order n=Δ(G)(γt(G)-1)+1. 相似文献
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Roe algebras are C?-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d). In the special case that X=Γ is a finitely generated group and d is a word metric, the simplest Roe algebra associated to (Γ,d) is isomorphic to the crossed product C?-algebra l∞(Γ)?rΓ. 相似文献
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Let F be an infinite field with characteristic not equal to two. For a graph G=(V,E) with V={1,…,n}, let S(G;F) be the set of all symmetric n×n matrices A=[ai,j] over F with ai,j≠0, i≠j if and only if ij∈E. We show that if G is the complement of a partial k -tree and m?k+2, then for all nonsingular symmetric m×m matrices K over F, there exists an m×n matrix U such that UTKU∈S(G;F). As a corollary we obtain that, if k+2?m?n and G is the complement of a partial k-tree, then for any two nonnegative integers p and q with p+q=m, there exists a matrix in S(G;R) with p positive and q negative eigenvalues. 相似文献
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Given n independent standard normal random variables, it is well known that their maxima Mn can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance dn between the normalized Mn and its associated limit distribution is less than 3/log?n. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with dn≤C(m)/log?n for n≥m≥5. Furthermore, the function C(m) is computed explicitly, which satisfies C(m)≤1 and limm→∞?C(m)=1/3. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function. 相似文献
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Let M=(mij) be a nonnegative irreducible n×n matrix with diagonal entries 0. The largest eigenvalue of M is called the spectral radius of the matrix M , denoted by ρ(M). In this paper, we give two sharp upper bounds of the spectral radius of matrix M. As corollaries, we give two sharp upper bounds of the distance matrix of a graph. 相似文献
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In this paper, we consider the problem (Pε) : Δ2u=un+4/n-4+εu,u>0 in Ω,u=Δu=0 on ∂Ω, where Ω is a bounded and smooth domain in Rn,n>8 and ε>0. We analyze the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev inequality as ε→0 and we prove existence of solutions to (Pε) which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε small, (Pε) has at least as many solutions as the Ljusternik–Schnirelman category of Ω. 相似文献
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Let S(Gσ) be the skew adjacency matrix of the oriented graph Gσ of order n and λ1,λ2,…,λn be all eigenvalues of S(Gσ). The skew spectral radius ρs(Gσ) of Gσ is defined as max{|λ1|,|λ2|,…,|λn|}. In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2. 相似文献
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It is proved that for each prime field GF(p), there is an integer np such that a 4-connected matroid has at most np inequivalent representations over GF(p). We also prove a stronger theorem that obtains the same conclusion for matroids satisfying a connectivity condition, intermediate between 3-connectivity and 4-connectivity that we term “k-coherence”. 相似文献
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For a function f(x) that is smooth on the interval x∈[a,b] but otherwise arbitrary, the real-valued roots on the interval can always be found by the following two-part procedure. First, expand f(x) as a Chebyshev polynomial series on the interval and truncate for sufficiently large N. Second, find the zeros of the truncated Chebyshev series. The roots of an arbitrary polynomial of degree N , when written in the form of a truncated Chebyshev series, are the eigenvalues of an N×N matrix whose elements are simple, explicit functions of the coefficients of the Chebyshev series. This matrix is a generalization of the Frobenius companion matrix. We show by experimenting with random polynomials, Wilkinson's notoriously ill-conditioned polynomial, and polynomials with high-order roots that the Chebyshev companion matrix method is remarkably accurate for finding zeros on the target interval, yielding roots close to full machine precision. We also show that it is easy and cheap to apply Newton's iteration directly to the Chebyshev series so as to refine the roots to full machine precision, using the companion matrix eigenvalues as the starting point. Lastly, we derive a couple of theorems. The first shows that simple roots are stable under small perturbations of magnitude ε to a Chebyshev coefficient: the shift in the root x* is bounded by ε/df/dx(x*)+O(ε2) for sufficiently small ε. Second, we show that polynomials with definite parity (only even or only odd powers of x) can be solved by a companion matrix whose size is one less than the number of nonzero coefficients, a vast cost-saving. 相似文献
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Let K be a closed convex subset of a q-uniformly smooth separable Banach space, T:K→K a strictly pseudocontractive mapping, and f:K→K an L-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1), let xt be the unique fixed point of tf+(1-t)T. We prove that if T has a fixed point, then {xt} converges to a fixed point of T as t approaches to 0. 相似文献
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