Abstract
Dendrimers are hyper-branched macromolecules having various applications in diverse fields like supra-molecular chemistry, drug delivery and nanotechnology etc. The certain graph invariants such as dominating number and power dominating number can be used to characterize large number of physical properties like physio-chemical properties, thermodynamic properties, chemical and biological activities, etc. A subset of a simple undirected graph is called dominating set if every vertex of the given graph is either in that set or is adjacent to some vertex in that set. The minimum number of elements in that kind of set is called domination number. A subset of the vertex set of a graph G is said to be power dominating set (PDS) of G, if every vertex and every edge in G is observed by P. The minimum cardinality of P of a graph G is called power domination number. In this paper, the domination number and power domination number of some nanostars dendrimers have been determined.
Highlights
A dendrimer is a molecule that is manufactured artificially and it has well defined chemical structure
The structure of dendrimers are composed by three major architectural components: one component is core which is the basic component in construction of dendrimer, branches which added in each step recursively to create a tree like structure and the end groups
The nanostar dendrimer behaves as macroparticles which appear to be photon funnels and is like artificial antennas
Summary
A dendrimer is a molecule that is manufactured artificially and it has well defined chemical structure. A subset P of the vertex set is said to be a power dominating set (abbreviated as PDS) if every vertex and every edge in G is observed by P. A PDS of G with the minimum cardinality is called power domination number and is denoted by γP(G). The Phase Measurement Unit (PMU ) is used to measure the voltage of node and current phase of the edges connected to the node in electric power network. A chemical graph is a labeled graph whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds Many chemical structures such as Sierpinski networks [13], silicate networks [10] were modeled as graphs and studied. We study the domination and power domination for complex chemical networks like some infinite families of Nanostar Dendrimers. We derive the exact values of domination and power domination for these classes of complex chemical networks
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