On the divisibility of H-shape trees and their spectral determination

Abstract

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 P2,2;n−42,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≠10,13,15.