Let B be a weighted generalized Bethe tree of k levels (k > 1) in which nj is the number of vertices at the level k− j+1 (1 ≤ j ≤ k). Let ∆ ⊆ {1, 2, . . . , k − 1} and F= {Gj : j ∈ ∆}, where Gj is a prescribed weighted graph on each set of children of B at the level k−j+1. In this paper, the eigenvalues of a block symmetric tridiagonal matrix of order n1 +n2 + · · ·+nk are characterized as the ...

Limit points for the positive eigenvalues of the normalized Laplacian matrix of a graph are considered. Specifically, it is shown that the set of limit points for the j-th smallest such eigenvalues is equal to [0, 1], while the set of limit points for the j-th largest such eigenvalues is equal to [1, 2]. Limit points for certain functions of the eigenvalues, motivated by considerations for rand...