Absolute algebraic connectivity of trees

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Abstract
The absolute algebraic connectivity of a graph G is the maximum of algebraic connectivities (second smallest eigenvalue of the Laplacian) for all nonnegative valuations of G whose average value on the edges of G is one. We prove that for a tree T-(V.E). this number is equal to the reciprocal of the variance of T which is the number where d(p,q) means the distance of (metric) points of the tree and M the absolute center of gravity of T, i.e. the (metric)point whose sum of squares of the distances to the vertices is minimal.(Metric means that points within the edges are also allowed). The absolute algebraic connectivity of a tree is always a rational number.

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