Theoretical Statistics and Mathematics Unit, ISI Delhi

January 24, 2018 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Fouzul Atik,
ISI Delhi

Title:
On distance and Laplacian matrices of trees with matrix weights

Abstract of Talk

The \emph{distance matrix} of a simple connected graph $G$ is
$D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the vertices
$i$ and $j$ in $G$. We consider a weighted tree $T$ on $n$ vertices
with edge weights are square matrix of same size. The distance
$d_{ij}$ between the vertices $i$ and $j$ is the sum of the weight
matrices of the edges in the unique path from $i$ to $j$. We
establish a characterization for the trees in terms of rank of
(matrix) weighted Laplacian matrix associated with it. We present a
necessary and sufficient condition for the distance matrix $D$, with
matrix weights, to be invertible and the formula for the inverse of
$D$, if it exists. Then we study some of the properties of the
distance matrices of matrix weighted trees in connection with the
Laplacian matrices, incidence matrices and g-inverses. Finally, we
derive an interlacing inequality of the eigenvalues of distance and
Laplacian matrices for the case of positive definite matrix weights.