Previous |  Up |  Next


A tree is classified as being type I provided that there are two or more Perron branches at its characteristic vertex. The question arises as to how one might construct such a tree in which the Perron branches at the characteristic vertex are not isomorphic. Motivated by an example of Grone and Merris, we produce a large class of such trees, and show how to construct others from them. We also investigate some of the properties of a subclass of these trees. Throughout, we exploit connections between characteristic vertices, algebraic connectivity, and Perron values of certain positive matrices associated with the tree.
[1] M. Fiedler: Algebraic connectivity of graphs. Czechoslovak Math. J. 23 (98) (1973), 298–305. MR 0318007 | Zbl 0265.05119
[2] M. Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory. Czechoslovak Math. J. 25 (100) (1975), 619–633. MR 0387321
[3] R. Grone and R. Merris: Algebraic connectivity of trees. Czechoslovak Math. J. 37 (112) (1987), 660–670. MR 0913997
[4] S. Kirkland, M. Neumann and B. Shader: Characteristic vertices of weighted trees via Perron values. Linear and Multilinear Algebra 40 (1996), 311–325. DOI 10.1080/03081089608818448 | MR 1384650
[5] R. Merris: Characteristic vertices of trees. Linear and Multilinear Algebra 22 (1987), 115–131. DOI 10.1080/03081088708817827 | MR 0936566 | Zbl 0636.05021
[6] R. Merris: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197/198 (1994), 143–176. MR 1275613 | Zbl 0802.05053
Partner of
EuDML logo