Geodesic
Spectra of weighted compound graphs of generalized Bethe trees
The electronic journal of linear algebra, Tome 18 (2009), pp. 30-57.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let Gm be a connected weighted graph on m vertices. Let B i : $1 \leq i \leq m$ be a set oftrees such that, for i = 1, 2, . . . , m, (i) B i is a generalized Bethe tree of k i levels, (ii) the vertices of B i at the level j have degree d i,k i - j+1 for j = 1, 2, . . . , k i , and (iii) the edges of B i joining the vertices at the level j with the vertices at the level (j + 1) have weight w i,k i - j for j = 1, 2, . . . , k i - 1. Let Gm B i : $1 \leq i \leq m$ be the graph obtained from G m and the trees B1 , B 2 , . . . , B m by identifying the root vertex of Bi with the ith vertex of G m . A complete characterization is given ofthe eigenvalues ofthe Laplacian and adjacency matrices of Gm B i : $1 \leq i \leq m$ together with results about their multiplicities. Finally, these results are applied to the particular case $B1 = B2 = \cdot \cdot \cdot = B$ m .
Classification : 15A48
Keywords: weighted graph, generalized Bethe tree, Laplacian matrix, adjacency matrix, spectral radius, algebraic connectivity 
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     title = {Spectra of weighted compound graphs of generalized {Bethe} trees},
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Rojo, Oscar; Medina, Luis. Spectra of weighted compound graphs of generalized Bethe trees. The electronic journal of linear algebra, Tome 18 (2009), pp. 30-57. http://geodesic.mathdoc.fr/item/ELA_2009__18__a54/