Geodesic
Spectra of copies of Bethe trees attached to path and applications
Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 33 (2008) no. 1.

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The Bethe tree $B_{d,k}$ is the rooted tree of $k$ levels whose root vertex has degree $d$ , the vertices from level 2 to level $k-1$ have degree $d+1$ , and the vertices at level $k$ have degree 1. This paper gives a decomposition of the characteristic polynomial of the adacency matrix of the tree $T(d,k,r)$ , obtained by attaching copies of $B(d,k)$ to the vertices of the $r$-vertex path. Moreover, lower and upper bounds for the energy of $T(d,k,r)$ are obtained. 
@article{BASS_2008_33_1_a3,
     author = {Mar{\'\i}a Robbiano and I. Gutman and R. Jim\'enez and B. San Martin},
     title = {Spectra of copies of {Bethe} trees attached to path and applications},
     journal = {Bulletin de l'Acad\'emie serbe des sciences. Classe des sciences math\'ematiques et naturelles},
     pages = {59 - 81},
     publisher = {mathdoc},
     volume = {33},
     number = {1},
     year = {2008},
     zbl = {1199.05240},
     url = {http://geodesic.mathdoc.fr/item/BASS_2008_33_1_a3/}
}
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María Robbiano; I. Gutman; R. Jiménez; B. San Martin. Spectra of copies of Bethe trees attached to path and applications. Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 33 (2008) no. 1. http://geodesic.mathdoc.fr/item/BASS_2008_33_1_a3/