Spectra of copies of Bethe trees attached to path and applications. (English) Zbl 1199.05240

Authors’ abstract: 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 adjacency 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.


05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C05 Trees
05C31 Graph polynomials
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