Constructing Equienergetic Graphs Using Windmill Graph

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 K2,2,...,2. Its adjacency spectrum, together with the isolated vertex, is {(2n − 2)1, (−2)n−1, 0n+1}, yielding energy 4(n− 1). Since K2n−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.