The cospectral equivalence classes of graphs having an isolated vertex

AbstractLet ϕ(G,λ) be the characteristic polynomial of a graph G. Two graphs G and H are cospectral, denoted by G∼H, if ϕ(G,λ)=ϕ(H,λ). By [G]ϕ we denote the cospectral equivalence class determined by G under “∼”. A graph G is said to be determined by its spectrum (or simply G is a DS-graph) if H≅G whenever H∼G. In this paper, we determine the cospectral equivalence classes of three kinds of graphs having an isolated vertex, find several DS-graphs and identify the graph that has the fourth minimum index among all connected graphs with n vertices