Graham et al. [3] proved a very attractive theorem about the determinant of the distance matrix DG of a connected graph G as a function of the distance matrix of its 2-connected blocks. In a connected graph, the distance between two vertices d(u, v) is the length of the shortest path between them. Let A be an n n matrix. Recall that for 1 i, j n, the cofactor ci,j is defined as ( 1) iþj times t...

Network Coordinate Systems (NCS) are promising techniques to predict unknown network distances from a limited number of measurements. Most NCS algorithms are based on metric space embedding and suffer from the inability to represent distance asymmetries and Triangle Inequality Violations (TIVs). To overcome these drawbacks, we formulate the problem of network distance prediction as guessing the...

The problem of computing the distance in the Frobenius norm of a given real irreducible tridiagonal matrix T to the algebraic variety of real normal irreducible tridiagonal matrices is solved. Simple formulas for computing the distance and a normal tridiagonal matrix at this distance are presented. The special case of tridiagonal Toeplitz matrices also is considered.

To classify time series by nearest neighbors, we need to specify or learn one or several distance measures. We consider variations of the Mahalanobis distance measures which rely on the inverse covariance matrix of the data. Unfortunately — for time series data — the covariance matrix has often low rank. To alleviate this problem we can either use a pseudoinverse, covariance shrinking or limit ...

The spectrum of a graph has been widely used in graph theory to characterise the properties of a graph and extract information from its structure. It has been less popular as a representation for pattern matching for two reasons. Firstly, more than one graph may share the same spectrum. It is well known, for example, that very few trees can be uniquely specified by their spectrum. Secondly, the...

A comparative study of 36 molecular descriptors derived from the topological distance matrix and van der Waals space is carried out within this paper. They are partitioned into 16 generalized topological distance matrix indices, 11 topological distance indices known in the literature (seven obtained from eigenvalues/eigenvectors of distance matrix), and 9 van der Waals molecular descriptors. Th...

A distance-based method to reconstruct a phylogenetic tree with n leaves takes a distance matrix, n × n symmetric matrix with 0s in the diagonal, as its input and reconstructs a tree with n leaves using tools in combinatorics. A safety radius is a radius from a tree metric (a distance matrix realizing a true tree) within which the input distance matrices must all lie in order to satisfy a preci...

A computer program is developed to compute distance polynomials of graphs containing up to 200 vertices. The code also computes the eigenvalues and the eigenvectors of the distance matrix. It requires as input only the neighborhood information from which the program constructs the distance matrix. The eigenvalues and eigenvectors are computed using the Givens-Householder method while the charac...

In information retrieval, documents(or images) are represented as vectors and the whole repository is represented as a matrix. Some similarity, distance and norm measurements between documents(or images) involve matrix computation. The challenge is the matrix may be too large to store, and compute. The idea is to reduce the matrix size while at the same time preserves characteristics such as Eu...

The aim of this note is to establish new spectral bounds for the harmonic matrix. The Harary matrix of given a connected graph G of order n, say RD(G), is an n-by-n symmetric matrix, such that (RD(G))ij =  1 dij , if i < j, 0 , if i = j, where dij denotes the distance between the vertices i and j [10, 11]. This matrix (originally known as reciprocal distance matrix [11]) is particulary well-k...