Some results on graphs with exactly two main eigenvalues

where ni = nβ 2 i (i = 1, 2); β1 and β2 denote the main angles of μ1 and μ2, respectively. Further, let G be any connected or disconnected graph (not necessarily with two main eigenvalues). Let S be any subset of the vertex set V (G) and let GS be the graph obtained from the graph G by adding a new vertex x which is adjacent exactly to the vertices from S. If σ(GS1) = σ(GS2) then we prove that σ(GS1) = σ(GS2) if and only if σ(GT1) = σ(GT2), where σ(G) is the spectrum of G and Ti = V (G) \ Si for i = 1, 2. However, if G is a connected graph which has exactly two main eigenvalues we prove the following results: (i) if σ(GS1) = σ(GS2) then σ(GS1) = σ(GS2); (ii) if σ(G ) = σ(G) then σ(Gi) = σ(Gj), where G = G \ k denotes the corresponding vertex deleted subgraph of G.