On Spectra Of Variants Of The Corona Of Two Graphs And Some New Equienergetic Graphs

Let G and H be two graphs. The join G ∨ H is the graph obtained by joining every vertex of G with every vertex of H. The corona G ○ H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the i-th vertex of G to every vertex in the i-th copy of H. The neighborhood corona G★H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the neighbors of the i-th vertex of G to every vertex in the i-th copy of H. The edge corona G ◇ H is the graph obtained by taking one copy of G and |E(G)| copies of H and joining each terminal vertex of i-th edge of G to every vertex in the i-th copy of H. Let G₁, G₂, G₃ and G₄ be regular graphs with disjoint vertex sets. In this paper we compute the spectrum of (G₁ ∨ G₂) ∪ (G1 ★ G₃), (G₁ ∨ G₂) ∪ (G₂ ★ G₃) ∪ (G₁ ★ G₄), (G₁ ∨ G₂) ∪ (G₁ ○ G₃), (G₁ ∨ G₂) ∪ (G₂ ○ G₃) ∪ (G₁ ○ G₄), (G₁ ∨ G₂) ∪ (G₁ ◇ G₃), (G₁ ∨ G₂) ∪ (G₂ ◇ G₃) ∪ (G₁ ◇ G₄), (G₁ ∨ G₂) ∪ (G₂ ○ G₃) ∪ (G₁ ★ G₃), (G₁ ∨ G₂) ∪ (G₂ ○ G₃) ∪ (G₁ ◇ G₄) and (G₁ ∨ G₂) ∪ (G₂ ★ G₃) ∪ (G₁ ◇ G₄). As an application, we show that there exist some new pairs of equienergetic graphs on n vertices for all n ≥ 11.