  A graph $G$ is divisible by a graph $H$ if the characteristic polynomial of $G$ is divisible by that of $H$. In this paper, a necessary and sufficient condition for recursive graphs to be divisible by a path is used to show that the H-shape graph $P_{2,2;n-4}^{2,n-7}$, known to be (for $n$ large enough) the minimizer of the spectral radius among the graphs of order $n$ and diameter $n-5$, is determined by its adjacency spectrum if and only if $n \neq 10,13,15$. 
On the divisibility of H-shape trees and their spectral determination