Kragujevac J. Math. 26 (2004) 19-24.

LAPLACIAN MATRIX AND DISTANCE IN TREES

Damir Vukicevic1 and Ivan Gutman2

1Department of Mathematics, University of Split, Croatia
2Faculty of Science, P. O. Box 60, 34000 Kragujevac, Serbia & Montenegro

Abstract. We provide a simple proof of an expression for the distance between vertices of a tree T in terms of the Laplacian matrix L of this tree. Let vi and vj be distinct vertices of T, at distance d(vi,vj). Then d(vi,vj) = det L[i,j], where L[i,j] is the submatrix, obtained by deleting the i-th and the j-th rows and columns from L.