Asian Research Journal of Mathematics

This study develops constructions of equienergetic graphs from a windmill graph and selected Cartesian products. Graph energy is taken as the sum of the absolute values of the adjacency eigenvalues, and two non-isomorphic graphs are equienergetic when these sums coincide. For the windmill graph W(3, n), formed from n copies of K₃ sharing a common vertex, the graph obtained by joining vertices whose distance in W(3, n) is exactly two is analysed. The common vertex becomes isolated, while the remaining vertices induce the complete n-partite graph K_2,2,...,2. Its adjacency spectrum, together with the isolated vertex, is {(2n − 2)₁, (−2)ⁿ⁻¹, 0ⁿ⁺¹}, yielding energy 4(n− 1). Since K_2n−1 has spectrum {(2n − 2)¹, (−1)²ⁿ⁻²} and the same energy, the two graphs are equienergetic. The manuscript also records that the energy of the shadow graph S(G) is twice the energy of G and uses the energies of complete and star graphs to identify another equienergetic family. Finally, Cartesian-product examples are considered. The spectrum of C₃&K₂ gives energy 8, equal to that of K₅; corresponding calculations give energies 12 and 16 for C₄&K₂ and C₆&K₂, matching K₇ and K₉, respectively. These constructions illustrate how spectral decomposition can be used to obtain explicit equienergetic graph pairs.