Proof of a Conjecture of Graham and Lovasz concerning Unimodality of Coefficients of the Distance Characteristic Polynomial of a Tree

The conjecture of Graham and Lov ́asz that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal is proved; it is also shown that the (normalized) coefficients are log-concave. Upper and lower bounds on the location of the peak are established.