It is known from algebraic graph theory that if L the Laplacian matrix of some tree G with a vertex degree sequence d=(δ1,…,δn)⊤ and D its distance matrix, then LD+2I=(2⋅1−d)1⊤, where 1 an all-ones column vector. We prove converse proposition: this identity holds for d D, essentially tree, matrix. This result immediately generalizes to weighted graphs. Therefore, above bilinear equation in L, c...

this study is a step toward identifying spatial and temporal patterns of precipitation on the territory of iraq. the purpose of this analysis, the data related to the network database aphrodite a 57 -year period (2007-1951), was used. the data matrix consists of three cell location, the original data (rainfall) and the calendar matrix .surfer was drawn up. also analyzed for the first time and t...

R. B. Bapat 2 A. K. Lal3 Sukanta Pati 4 Abstract We consider a q-analogue of the distance matrix (called the q-distance matrix) of an unweighted tree and give formulae for the inverse and the determinant, which generalize the existing formulae for the distance matrix. We obtain the Smith normal form of the q-distance matrix of a tree. The relationship of the q-distance matrix with the Laplacian...

Applying weighted network measures to distance matrices Many approaches to the analysis of weighted networks are not designed for fully connected weighted networks. However, as any distance matrix between objects is a fully connected weighted network, such networks are extremely common. In earlier work we derived an approach for the analysis of weighted networks which also works on fully connec...

it is well known that the matrix exponential function has practical applications in engineering and applied sciences. in this paper, we present some new explicit identities to the exponential functions of a special class of matrices that are known as central-symmetric $x$-form. for instance, $e^{mathbf{a}t}$, $t^{mathbf{a}}$ and $a^{mathbf{a}t}$ will be evaluated by the new formulas in this par...