Distance equienergetic graphs of diameter 4

Let ${\Gamma }_1$, ${\Gamma }_2$ and ${\Gamma }_3$ be graphs with pairwise disjoint vertex sets. The graph $\Theta ({\Gamma }_1,{\Gamma }_2,{\Gamma }_3)$ is obtained from the graphs ${\Gamma }_1\circ {\Gamma }_3$ (the corona product) and ${\Gamma }_2$ by joining each vertices of ${\Gamma }_1$ in ${\Gamma }_1\circ {\Gamma }_3$ with every vertices in ${\Gamma }_2$. 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 $\Theta ({\Gamma }_1,{\Gamma }_2,{\Gamma }_3)$ 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 ($n\ge 6$) which are both adjacency equienergetic and distance equienergetic graphs.