Abstract
A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let P m be a path of m vertices. Let { B i : 1 ⩽ i ⩽ m } be a set of generalized Bethe trees. Let P m { B i : 1 ⩽ i ⩽ m } be the tree obtained from P m and the trees B 1 , B 2 , … , B m by identifying the root vertex of B i with the i - th vertex of P m . We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of P m { B i : 1 ⩽ i ⩽ m } . In particular, we characterize their spectral radii and the algebraic conectivity. Moreover, we derive results concerning their multiplicities. Finally, we apply the results to the case B 1 = B 2 = … = B m .
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