Let V be a real vector space of dimension d and V ∗ its dual space. By a cone in V ∗ we will always mean a closed polyhedral cone σ with apex 0 such that σ ∩ −σ = {0}. Let Σ be a fan in V ∗, i.e., a collection of cones such that (1) if σ ∈ Σ then any face of σ belongs to Σ, (2) if σ1, σ2 ∈ Σ then σ1 ∩ σ2 is a face in both. We will assume that Σ is complete, that is ∪Σ = V ∗. The elements of Σ a...

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 ...