Energy of a set of vertices in a Graph

Given a finite graph G = (V,E), and any proper subset D of the vertex set V := V (G) of G, we associate a nonnegative integral matrix AD(G) = (aij) of order |D| ◊ |D| with D so that the i th diagonal entry in the matrix counts precisely the number of edges that join the i th vertex of D with vertices in V D so that these partial degrees of the vertices in D are precisely the eigenvalues of AD(G) whence their sum may be conceived as the energy "D(G) of the given set D. Invoking the underlying notion of incidence matrix of D, we introduce in this paper the notion of robust domination energy (or, rd-energy) and shear domination energy (or, sd-energy) of G as the maximum (minimum, respectively) energy of a minimal dominating set in G. We raise several interesting open problems and connections of these notions with other well known ones in graph theory.