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Summary:
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.
References:
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[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
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