Constructing pairs of equienergetic and non-cospectral graphs

The energy   of a simple graph GG is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two graphs of the same order are said to be equienergetic   if they have the same energy. Several ways to construct equienergetic non-cospectral graphs of very large size can be found in the literature. The aim of this work is to construct equienergetic non-cospectral graphs of small size. In this way, we first construct several special families of such graphs, using the product and the cartesian product of complete graphs. Afterwards, we show how one can obtain new pairs of equienergetic non-cospectral graphs from the starting ones. More specifically, we characterize the connected graphs GG for which the product and the cartesian product of GG and K2K2 are equienergetic non-cospectral graphs and we extend Balakrishnan’s result: For a non-trivial graph  GG, G⊗C4G⊗C4and  G⊗K2⊗K2G⊗K2⊗K2are equienergetic non-cospectral graphs, given in [R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287–295].