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A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ≥ 3 and n ≥ 6 be given. Let Pd−1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. Let p = [p1, p2, ..., pd−1] such that p1 ≥ 1,... more
A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ≥ 3 and n ≥ 6 be given. Let Pd−1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. Let p = [p1, p2, ..., pd−1] such that p1 ≥ 1, p2 ≥ 1, ..., pd−1 ≥ 1. Let C (p) be the caterpillar obtained from the stars Sp1 , Sp2 , ...,Spd−1 and the path Pd−1 by identifying the root of Spi with the i−vertex of Pd−1. Let n > 2 (d − 1) be given. Let C = {C (p) : p1 + p2 + ... + pd−1 = n − d + 1} and S = {C(p) ∈ C : pj = pd−j , j = 1, 2, · · · , ⌊ d − 1 2 ⌋}. In this paper, the caterpillars in C and in S having the maximum and the minimum algebraic connectivity are found. Moreover, the algebraic connectivity of a caterpillar in S as the smallest eigenvalue of a 2 × 2 - block tridiagonal matrix of order 2s × 2s if d = 2s + 1 or d = 2s + 2 is characterized. Work supported by CNPq 300563/94-9, Brazil.
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A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let Pm be a path of m vertices. Let {Bi:1⩽i⩽m} be a set of generalized Bethe trees. Let Pm{Bi:1⩽i⩽m} be the tree obtained... more
A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let Pm be a path of m vertices. Let {Bi:1⩽i⩽m} be a set of generalized Bethe trees. Let Pm{Bi:1⩽i⩽m} be the tree obtained from Pm and the trees B1,B2,…,Bm by identifying the root vertex of Bi with the
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The level of a vertex in a rooted graph is one more than its distance from the root vertex. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. We characterize completely the eigenvalues of... more
The level of a vertex in a rooted graph is one more than its distance from the root vertex. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. We characterize completely the eigenvalues of the Laplacian, signless Laplacian and adjacency matrices of a weighted rooted graph G obtained from a weighted generalized Bethe tree of k levels and weighted cliques in which(1)the edges connecting vertices at consecutive levels have the same weight,(2)each set of children, in one or more levels, defines a weighted clique, and(3)cliques at the same level are isomorphic.These eigenvalues are the eigenvalues of symmetric tridiagonal matrices of order j×j,1⩽j⩽k. Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity. Finally, we apply the results to the unweighted case and some particular graphs are studied.