On Weighted Zagreb Energies For Certain Classes Of Graphs

This work presents matrix formulations corresponding to the weighted first and second Zagreb indices. Building on these matrices, we define the associated weighted first and second Zagreb energies and derive explicit expressions of these energies for regular graphs, including the cycle, complete, complete bipartite, and complete tripartite graphs. Using these expressions, we identify families of non-isomorphic weighted-Zagreb-equienergetic graphs, and we classify graphs into hyper-energetic, non-hyper-energetic, and border-energetic types with respect to both weighted Zagreb energies. In addition, the weighted first and second Zagreb energies of the q-splitting graph of the cycle, complete, complete bipartite, and complete tripartite graphs are derived, in terms of parameters of underlying graph, and the resulting spectral relationships are checked for internal consistency.