Given a tree T, let q(T) be the minimum number of distinct eigenvalues in symmetric matrix whose underlying graph is T. It well known that q(T)≥d(T)+1, where d(T) diameter and T said to diminimal if q(T)=d(T)+1. In this paper, we present families trees any fixed diameter. Our proof constructive, allowing us compute, for d these families, M with spectrum has exactly d+1 eigenvalues.

Given a compact connected set E in the unit disk B2, we give new upper bound for conformal capacity of condenser (B2,E) terms hyperbolic diameter t E. Moreover, t>0, construct and apply novel numerical methods to show that it has larger than with same diameter. The is called Reuleaux triangle geometry constant width equal t.