Topological indices (TIs) transform a molecular graph into a number. The TIs are a vital tool for quantitative structure activity relationship (QSAR) and quantity structure property relationship (QSPR). In this paper, we constructed two classes of Benes network: horizontal cylindrical Benes network and vertical cylindrical Benes network obtained by identification of vertices of first rows with last row and first column with last column of Benes network, respectively. We derive analytical close formulas for general Randić connectivity index, general Zagreb, first and the second Zagreb (and multiplicative Zagreb), general sum connectivity, atom-bond connectivity (), and geometric arithmetic index of the two classes of Benes networks. Also, the fourth version of and the fifth version of indices are computed for these classes of networks.

In a molecular graph, molecules are associated with some numerical values these values are known as topological indices. From the M-polynomial of molecular structure we can derived degree based topological indices. We can derived chemical and physical properties of chemical compound from the topological indices. To find the strain energy, melting point, boiling point, distortion and stability of chemical compound usually mathematician used topological indices. Moreover topological indices also make relation between biological activities of compound with physical properties. In this paper, we determined the M-polynomials of the structure of the molecules of polyphenylene nanotube and nanotori. Then we derived some closed formulas for well-known topological indices, first Zagreb index, second Zagreb index, second Modified Zagreb index $m{M}_2(G),$ general Randić index, Symmetric division index, Harmonic index, inverse sum index for polyphenylene nanotube structure of molecules.

We investigate the vertex total and edge total modication of the well-known irregularity strength of graphs. We have determined the exact values of the total vertex irregularity strength and the total edge irregularity strength of a disjoint union of friendship graphs.

Counting polynomials are mathematical functions having as coefficients the multiplicity of property partition and as exponent the extent of the corresponding partition. In this work, four counting polynomials are computed: omega, theta, sadhana, and Padmakar–Ivan (PI) polynomials of benzene ring embedded in the P-type surface on 2-D network and n-phenylene. Moreover, the closed formulas of the corresponding indices of these polynomials are also proposed. These results could help to better understand the topology of these important structures.