Graph Indices for Cartesian Product of -sum of Connected Graphs.

BACKGROUND: A topological index is a real number associated with a graph that provides information about its physical and chemical properties and their correlations. Topological indices are being used successfully in Chemistry, Computer Science, and many other fields. METHODS: In this article, we apply the well-known Cartesian product on F-sums of connected and finite graphs. We formulate sharp limits for some famous degree-dependent indices. RESULTS: Zagreb indices for the graph operations T(G), Q(G), S(G), R(G), and their F-sums have been computed. By using orders and sizes of component graphs, we derive bounds for Zagreb indices, F-index, and Narumi-Katayana index. CONCLUSION: The formulation of expressions for the complicated products on F-sums, in terms of simple parameters like maximum and minimum degrees of basic graphs, reduces the computational complexities.

Topological Properties of Sierpinski Network and its Application.

BACKGROUND: Sierpinski graphs S(n, k) are largely studied because of their fractal nature with applications in topology, chemistry, mathematics of Tower of Hanoi and computer sciences. Applications of molecular structure descriptors are a standard procedure which are used to correlate the biological activity of molecules with their chemical structures, and thus can be helpful in the field of pharmacology. OBJECTIVE: The aim of this article is to establish analytically closed computing formulae for eccentricity-based descriptors of Sierpinski networks and their regularizations. These computing formulae are useful to determine a large number of properties like thermodynamic properties, physicochemical properties, chemical and biological activity of chemical graphs Methods: At first, vertex sets have been partitioned on the basis of their degrees, eccentricities and frequencies of occurrence. Then these partitions are used to compute the eccentricity-based indices with the aid of some combinatorics. RESULTS: The total eccentric index and eccentric-connectivity index have been computed. We also compute some eccentricity-based Zagreb indices of the Sierpinski networks. Moreover, a comparison has also been presented in the form of graphs. CONCLUSION: These computations will help the readers to estimate the thermodynamic properties and physicochemical properties of chemical structure which are of fractal nature and can not be dealt with easily. A 3D graphical representation is also presented to understand the dynamics of the aforementioned topological descriptors.

On the bounds of degree-based topological indices of the Cartesian product of F-sum of connected graphs.

Topological indices are the mathematical tools that correlate the chemical structure with various physical properties, chemical reactivity or biological activity numerically. A topological index is a function having a set of graphs as its domain and a set of real numbers as its range. In QSAR/QSPR study, a prediction about the bioactivity of chemical compounds is made on the basis of physico-chemical properties and topological indices such as Zagreb, Randic and multiple Zagreb indices. In this paper, we determine the lower and upper bounds of Zagreb indices, the atom-bond connectivity (ABC) index, multiple Zagreb indices, the geometric-arithmetic (GA) index, the forgotten topological index and the Narumi-Katayama index for the Cartesian product of F-sum of connected graphs by using combinatorial inequalities.