The Minimum Monopoly Energy of a Graph

Abstract

In a graph $G(V,E)$, a subset $M \subseteq V(G)$ is called a monopoly set of $G$ if every vertex $v\in V - M$ has at least $\frac{d(v)}{2}$ neighbors in $M$. The monopoly size of $G$ is the minimum cardinality of a monopoly set among all monopoly sets in $G$, denoted by $mo(G)$. In this paper, we introduce minimum monopoly energy, denoted $E_M(G)$, of a graph $G$ and computed minimum monopoly energies of some standard graphs. Upper and lower bounds for $E_M(G)$ are established.