The cospectral equivalence classes of graphs having an isolated vertex

Let ϕ(G,λ)ϕ(G,λ) be the characteristic polynomial of a graph GG. Two graphs GG and HH are cospectral, denoted by G∼HG∼H, if ϕ(G,λ)=ϕ(H,λ)ϕ(G,λ)=ϕ(H,λ). By [G]ϕ[G]ϕ we denote the cospectral equivalence class determined by GG under “∼∼”. A graph GG is said to be determined by its spectrum (or simply GG is a DS-graph) if H≅GH≅G whenever H∼GH∼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 nn vertices.