In this paper, block distance matrices are introduced. Suppose F is a square block matrix in which each block is a symmetric matrix of some given order. If F is positive semidefinite, the block distance matrix D is defined as a matrix whose (i, j)-block is given by Dij = Fii+Fjj−2Fij . When each block in F is 1 × 1 (i.e., a real number), D is a usual Euclidean distance matrix. Many interesting ...

Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...

Let T be a tree on n vertices and let the n− 1 edges e1, e2, . . . , en−1 have weights that are s× s matrices W1,W2, . . . ,Wn−1, respectively. For two vertices i, j, let the unique ordered path between i and j be pi,j = er1er2 . . . erk . Define the distance between i and j as the s × s matrix Ei,j = ∏k p=1Wep . Consider the ns × ns matrix D whose i, j-th block is the matrix Ei,j . We give a f...