Spectra and Randic spectra of caterpillar graphs and applications to the energy

Abstract

Let $H$ be an undirected simple graph with vertices $v_{1},\ldots ,v_{k}$ and $G_{1},\ldots ,G_{k}$ be a sequence formed with $k$ disjoint graphs $G_{i}$, $i=1,\ldots ,k$. The $H$-generalized composition (or $H$% -join) of this sequence is denoted by $H\left[ G_{1},\ldots ,G_{k}\right] .$ 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.