Special ultrametric matrices and graphs

Special ultrametric matrices are, in a sense, extremal matrices in the boun dary of the set of ultrametric matrices introduced by Martinez, Michon, and San Martin [SIAM J. Matrix Anal. Appl., 15 (1994), pp. 98-106]. We show a simple construction of these matrices, if of order n, from nonnegatively ed ge-weighted trees on n vertices, or, equivalently, from nonnegatively edge- weighted paths. A general ultrametric matrix is then the sum of a nonnegati ve diagonal matrix and a special ultrametric matrix, with certain condition s fulfilled. The rank of a special ultrametric matrix is also recognized an d it is shown that its Moore-Penrose inverse is a generalized diagonally do minant M-matrix. Some results on the nonsymmetric case are included.