The minimum mean monopoly energy of a graph

The motivation for the study of the graph energy comes from chemistry, where the research on the so-called total pi - electron energy can be traced back until the 1930s. This graph invariant is very closely connected to a chemical quantity known as the total pi - electron energy of conjugated hydro carbon molecules. In recent times analogous energies are being considered, based on Eigen values of a variety of other graph matrices. In 1978, I.Gutman [1] defined energy mathematically for all graphs. Energy of graphs has many mathematical properties which are being investigated. The ordinary energy of an undirected simple finite graph G is defined as the sum of the absolute values of the Eigen values of its associated matrix. i.e. if mu(1), mu(2), ..., mu(n) are the Eigen values of adjacency matrix A(G), then energy of graph is Sigma(G) = Sigma(n)(i=1) vertical bar mu(i)vertical bar Laura Buggy, Amalia Culiuc, Katelyn Mccall and Duyguyen [9] introduced the more general M-energy or Mean Energy of G is then defined as E-M (G) = Sigma(n)(i=1)vertical bar mu(i) - (mu) over bar vertical bar, where (mu) over bar vertical bar is the average of mu(1), mu(2), ..., mu(n). A subset M subset of V (G), in a graph G (V, E), is called a monopoly set of G if every vertex v is an element of (V - M) has at least d(v)/2 neighbors in M. The minimum cardinality of a monopoly set among all monopoly sets in G is called the monopoly size of G, denoted by mo(G) Ahmed Mohammed Naji and N.D.Soner [7] introduced minimum monopoly energy E-MM [G] of a graph G. In this paper we are introducing the minimum mean monopoly energy, denoted by E-MM(M) (G), of a graph G and computed minimum monopoly energies of some standard graphs. Upper and lower bounds for E-MM(M) (G)are also established.Publisher's Versio