On spectra of variants of the corona of two graphs and some new equienergetic graphs

Abstract

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.