# Seminar at SMU 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.