Computing Topological Indices of Certain Networks

In mathematical chemistry, topological indices are molecular descriptors that are calculated on the molecular graph of a chemical compound. The molecular graph is a graph which is obtained from some chemical structures. The degree of every molecular graph cannot exceeds 4. Topological indices are numerical quantities of a graph that describe its topology. An atom represents a vertex and a bond between two atoms represents an edge in a molecular graph. Mainly there are three types of topological indices viz. , degree-based, distance based and eigenvalue-based topological indices. The first degree-based topological indices are the first and second Zagreb indices. The first Zagreb index M₁ is defined as the sum of squares of degrees of each vertex in a graph G and the second Zagreb index M₂ is the product of degree of every adjacent vertices. In this case the summation goes on the set of edges of a graph G. The most studied topological indices are degree-based topological indices. Motivated by these topological indices in this paper, we introduce five new degree-based topological indices based on the neighborhood degree of a vertex. Further, we compute the values of various nanostructures like hexagonal parallelogram P (m, n) nanotube, triangular benzenoid Gₙ, zigzag-edge coronoid fused with starphene nanotubes ZCS (k, l, m), dominating derived networks D₁, D₂, D₃, Porphyrin Dendrimer, Zinc-Porphyrin Dendrimer, Propyl Ether Imine Dendrimer, Poly (Ethylene amido amine Dendrimer, PAMAM dendrimers (? ? ? ? ? ? ? ? 1,? ? ? ? ? ? ? ? 2,? ? ? ? ? ? ? ? 1), linear polyomino chain Lₙ, Zₙ, Bₙ¹ (n≥3), Bₙ² (n≥3) and triangular, hourglass, and jagged-rectangle benzenoid systems of these indices. The standard computational techniques are used for the computation of topological indices of nanostructures. For the edge partition of the nanostructures the algebraic techniques are used. Using these techniques computation of topological indices became easy and also helped to get the more accurate results.