Distance equienergetic graphs of diameter 4

Let Γ₁, Γ₂ and Γ₃ be graphs with pairwise disjoint vertex sets. The graph Θ(Γ₁,Γ₂,Γ₃) is obtained from the graphs Γ₁∘Γ₃ (the corona product) and Γ₂ by joining each vertices of Γ₁ in Γ₁∘Γ₃ with every vertices in Γ₂. Two connected graphs are called distance equienergetic graphs if their distance energies are the same. Several methods for constructing distance equienergetic graphs have been presented in the literature, most constructed distance equienergetic graphs have diameters of 2 or 3. So the problem of constructing distance equienergetic graphs of diameter greater than 3 would be interesting. Another interesting problem posed by Indulal (2020) is to construct a pair of graphs which are both adjacency equienergetic and distance equienergetic. Motivated by these two problems, in this paper, we obtain the distance spectrum of Θ(Γ₁,Γ₂,Γ₃) when all these graphs are regular. As an application, we give a method to obtain distance equienergetic graphs of diameter 4. Also we construct a pair of graphs on 2n+1 vertices (n≥6) which are both adjacency equienergetic and distance equienergetic graphs.