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The 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. In this article we establish a characterization for the trees in terms of rank of (matrix) weighted Laplacian matrix associated with it. Then we establish 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. Also we study some of the properties of the distance matrices of matrix weighted trees in connection with the Laplacian matrices, g-inverses and eigenvalues.

The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the $i$th and $j$th vertices of $G$. The \emph{distance signless Laplacian matrix} of the graph $G$ is $D_Q(G)=D(G)+Tr(G)$, where $Tr(G)$ is a diagonal matrix whose $i$th diagonal entry is the transmission of the vertex $i$ in $G$. In this paper, first, upper and lower bounds for the spectral radius of a nonnegative matrix are constructed. Applying this result, upper and lower bounds for the distance and distance signless Laplacian spectral radius of graphs are given, and the extremal graphs for these bounds are obtained. Also, upper bounds for the modulus of all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius of graphs are given. These bounds are probably first of their kind as the authors do not find in the literature any bound for these eigenvalues. Finally, for some classes of graphs, it is shown that all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius lie in the smallest Ger\^sgorin disc of the distance (respectively, distance signless Laplacian) matrix.

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