The concept of local pseudo-distance-regularity, introduced in this paper, can be thought of as a natural generalization of distance-regularity for non-regular graphs. Intuitively speaking, such a concept is related to the regularity of graph 1 when it is seen from a given vertex. The price to be paid for speaking about a kind of distance-regularity in the non-regular case seems to be locality....

Let G be a simple connected graph. The terminal distance matrix of G is the distance matrix between all pendant vertices of G. In this paper, we introduce some general transformations that increase the terminal distance spectral radius of a connected graph and characterize the extremal trees with respect to the terminal distance spectral radius among all trees with a fixed number of pendant ver...

The QCD radiative corrections to the leptonic decay rate of the B c meson are calculated using the formalism of nonrelativistic QCD (NRQCD) to separate short-distance and long-distance effects. The B c decay constant is factored into a sum of NRQCD matrix elements each multiplied by a short-distance coefficient. The short-distance coefficient for the leading matrix element is calculated to orde...

A lot of machine learning algorithms are based on metric functions, which good functions lead to better results. Distance metric learning has been widely attracted by researchers in last decade. Kernel matrix is somehow a distance function which indicates the similarity between two instances in the feature space which contains high dimensions. Traditional distance metric learning approaches are...

W ABSTRACT e describe a bisection method to determine the 2-norm and Frobenius norm-g distance from a given matrix A to the nearest matrix with an eigenvalue on the ima inary axis. If A is stable in the sense that its eigenvalues lie in the open left half e plane, then this distance measures how "nearly unstable" A is. Each step provides ither a rigorous upper bound or a rigorous lower bound on...

In this note, we show how the determinant of the q-distance matrix Dq(T ) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85–88]. Further, by means of the result, w...

The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified alge...

Abstract Let T be a tree with n vertices and let D be the distance matrix of T. According to a classical result due to Graham and Pollack, the determinant of D is a function of n, but does not depend on T. We allow the edges of T to carry weights, which are square matrices of a fixed order. The distance matrix D of T is then defined in a natural way. We obtain a formula for the determinant of D...

In this note, we prove that all cop-win graphs G in the game in which the robber and the cop move at different speeds s and s′ with s′ < s, are δ-hyperbolic with δ = O(s). We also show that the dependency between δ and s is linear if s − s′ = Ω(s) and G obeys a slightly stronger condition. This solves an open question from the paper J. Chalopin et al., Cop and robber games when the robber can h...