Spectra and Randic Spectra of Caterpillar Graphs and Applications to the Energy - Andrade, Enide (2017)
Let H be an undirected simple graph with vertices v_1,...,v_k and G_1,...,G_k be a sequence formed with k disjoint graphs G_i,i=1,...,k. The H-generalized composition (or H-join) of this sequence is denoted by H[G_1,...,G_k]. In this work, we characterize the caterpillar graphs as a H-generalized composition and we study their spectra and Randi\'c spectra, respectively. As an application, we obtain an improved and tight upper bound for the Energy and the Randi\'c energy of these interesting trees.