The Minimum Monopoly Distance Energy of a Graph

In a graph G = (V,E), a set M ⊆ V is called a monopoly set of G if every vertex v ∈ V −M has at least d(v) 2 neighbors in M . The monopoly size mo(G) of G is the minimum cardinality of a monopoly set among all monopoly sets in G. In this paper, the minimum monopoly distance energy EMd(G) of a connected graphG is introduced and minimum monopoly distance energies of some standard graphs are computed. Some properties of the characteristic polynomial of the minimum monopoly distance matrix of G are obtained. Finally. Upper and lower bounds for EMd(G) are established.