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 ο 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 οG₃), (G₁ ∨ G₂)∪(G₂ οG₃)∪(G₁ οG₄), (G₁∨G₂)∪(G₁ ⋄G₃), (G₁ ∨ G₂) ∪ (G₂ ⋄ G₃) ∪ (G₁ ⋄ G₄), (G₁ ∨ G₂) ∪ (G₂ ο G₃) ∪ (G₁ ∗ G₃), (G₁ ∨ G₂) ∪ (G₂ ο 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.