Distance equienergetic graphs of diameter 4

Abstract

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.

Keywords

• Equienergetic graphs
• Distance equienergetic graphs
• Join graph
• Corona product