On eigenvalues and the energy of dendrimer trees

Abstract

A dendrimer tree Dn,k is a rooted tree with root v0 in which the root vertex v0 has k children, the vertices u satisfying the distance d(v0,u)=i have k−1 children for 1≤i≤n−1, and the vertices u such that d(v0,u)=n are pendant vertices. In this paper, we obtain almost all of eigenvalues of Dn,k except n+1 eigenvalues which are just the roots of the matching polynomial of a weighted path with n+1 vertices v0,v1,…,vn and n edges (v0,v1),(v1,v2),(v2,v3)⋯,(vn−1,vn), of weights equal to k,k−1,k−1,⋯,k−1, and we obtain a formula to compute the energy of Dn,k.