By a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice...

For an end $\tau$ and a tree $T$ of a graph $G$ we denote respectively by $m\left(\tau \right)$ and $m_T\left(\tau \right)$ the maximum numbers of pairwise disjoint rays of $G$ and $T$ belonging to $\tau$, and we define $\mathrm{t}m\left(\tau \right):=min\left\{m_T\left(\tau \right)T\text{is}\text{a}\text{spanning}\text{tree}\text{of}G\right\}$. In this paper we give partial answers—affirmative and negative ones—to the general problem of determining if, for a function $f$ mapping every end $\tau$ of $G$ to a cardinal $f\left(\tau \right)$ such that $\mathrm{t}m\left(\tau \right)\le f\left(\tau \right)\le m\left(\tau \right)$, there exists a spanning tree $T$ of $G$ such that $m_T\left(\tau \right)=f\left(\tau \right)$ for every end $\tau$ of $G$.