Abstract
The concept of Domination in graphs has application to the study of DNA structures. For investigating the chemical and physical properties, several topological indices used are Wiener index, Randic index, Zagreb index, Kier & Hall index that depends on vertex degree and distance sum, and have been used extensively for QSAR and QSPR studies.
 A Roman Dominating Function of G is function f: V→ {0, 1, 2} such that every vertex v for which f (v) = 0 has a neighbor u with f(u) = 2. The weight of a Roman dominating function f is w (f) = . The Roman domination number of a graph G is denoted by (G) and is the minimum weight of all possible Roman dominating functions. In this paper, we find Roman domination number of some chemicals graphs such as saturated hydrocarbons and unsaturated hydrocarbons, hexagonal chain, pyrene, Hexabenzocoronene, H-Phenylenic nanotube and N-Napthelenic nanotube.
Highlights
Graph theory has wide application in the study of the properties of molecules
Graph theory is used for the investigation of the properties through quantitative structure-activity relationships (QSARs) and QSPR models through graph invariants such as topological indices
[10] Suppose G is the molecular graph of chloroquine
Summary
Graph theory has wide application in the study of the properties of molecules. Graphs are used to represent chemical structures. A molecule consisting of atoms and atomic bound is correlated to the graph with vertices and edges where atom corresponds to vertex and atomic bound corresponds to edge and valance of atom corresponds to degree of vertex. This graphical representation reveals many chemical properties of molecules. The graph G with the function f : V→{ 0,1,2} supporting for each vertex v assigned f(v) = 0 has a neighbor u assigned f(u) = 2 and w(f) = ∑v∈V f(v) is called the weight of a Roman dominating function. Area is compared to vertex v is considered unstable if no armies are positioned there i.e f(v) = 0 and stable (f(v)∈{1,2}). In the HPhenylenic HPH[m, n] nanotube, m represents the number of hexagons (rows) in each column and n represents the number of hexagons(columns) in each row
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