-energy of graphs

Given a graph G = (V, E), with respect to a vertex partition we associate a matrix called -matrix and define the -energy, E (G) as the sum of -eigenvalues of -matrix of G. Apart from studying some properties of -matrix, its eigenvalues and obtaining bounds of -energy, we explore the robust(shear) -energy which is the maximum(minimum) value of -energy for some families of graphs. Further, we derive explicit formulas for E (G) of few classes of graphs with different vertex partitions