THE MINIMUM MEAN MONOPOLY ENERGY OF A GRAPH

THE MINIMUM MEAN MONOPOLY ENERGY OF A GRAPH

M. V. Chakradhara Rao, K. A. Venkatesh, D. V. Lakshmi

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Abstract

The motivation for the study of the graph energy comes from chemistry, where the research on the so-called total π - electron energy can be traced back until the 1930s. This graph invariant is very closely connected to a chemical quantity known as the total π - 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 val- ues of the Eigen values of its associated matrix. i.e. if μ1, μ2, ..., μn are the Eigen values of adjacency matrix A(G), then energy of graph is E(G) = ni=1 |μi|. Laura Buggy, Amalia Culiuc, Katelyn Mccall and Duyguyen [9] introduced the more general M-energy or Mean Energy of G is then defined as EM (G) = ni=1 |μi − μ ̄|, where μ ̄ is the average of μ1, μ2, ..., μn. A subset M ⊆ V (G), in a graph G (V, E), is called a monopoly set of G if every vertex v ∈ (V - M) has at least d(v) neighbors in M. The minimum cardinality of a monopoly set 2 among all monopoly sets in G is called the monopoly size of G, denoted by mo(G).Ahmed Mohammed NajiandN.D.Soner[7] introduced minimum monopoly energy EMM [G]ofa graph G. In this paper we are introducing the minimum mean monopoly energy, denoted by EM (G), of a graph G and computed minimum monopoly energies of some standard MM graphs. Upper and lower bounds for EM (G)are also established. MM

Keywords

Monopoly Set, Monopoly Size, Minimum Monopoly Matrix, Minimum Mo- nopoly Eigenvalues, Minimum Monopoly Energy and Minimum Mean Monopoly Energy of a graph