On Lanzhou and Ad-hoc Lanzhou Indices of Derived Graphs and Silicate Structures

Abstract

Degree-based topological indices are among the most widely used descriptors in chemical graph theory. These indices rely on the degrees (or valencies) of the vertices in a molecular graph, where the degree of a vertex corresponds to the number of edges (bonds) connected to it. One among these types of indices is the Lanzhou index, given by \( Lz(G) = \sum \limits _{a\in V(G)}d_G(a)^2d_{\overline{G}}(a),\) where \(d_G(a)\) and \(d_{\overline{G}}(a)\) denote the degree of the vertex a in G and the complement graph of G, respectively. Ad-hoc Lanzhou index, \(\overline{Lz}(G)\) is obtained by switching the roles of degrees of vertices, i.e., \(\overline{Lz}(G)=\sum \limits _{a \in V(G)}d_G(a)d_{\overline{G}}(a)^2.\) In this manuscript, expressions for the Lanzhou and Ad-hoc Lanzhou indices of derived graphs, namely, subdivision, line, vertex semi-total, edge semi-total, and total graphs, are obtained. Also, Lanzhou and Ad-hoc Lanzhou indices of \(Si_2C_3-\MakeUppercase {i}(s,t),\ Si_2C_3-\MakeUppercase {ii}(s,t),\) and \(Si_2C_3-\MakeUppercase {iii}(s,t)\) are obtained and their graphical analysis have been made.