A topological index is a structural descriptor of any molecule/nanostructure that characterizes its topology. In the QSAR and QSPR research, topological indices are employed to predict the physical characteristics associated with bioactivities and chemical reactivity within speci5c networks. 2D nanostructured materials have many exhibit numerous chemical, mechanical, and physical features. +ese nanomaterials are exceptionally thin, displaying high chemical functionality and anisotropy. For applications necessitating robust surface interactions on a small scale, 2D materials stand out as the optimal choice due to their expansive surface area and status as the thinnest among all discovered materials. +is paper characterized the neighborhood irregular topological invariants of nanostructures TUC4C8[p, q] and GTUC[p, q] and derived closed form expressions for them. A comparative analysis is then performed on the basis of these computed indices.

Chemical graph theory is a well-established discipline within chemistry that employs discrete mathematics to represent the physical and biological characteristics of chemical substances. In the realm of chemical compounds, graph theory-based topological indices are commonly employed to depict their geometric structure. The main aim of this paper is to investigate the degree-based topological indices of dominating David derived networks (DDDN) and assess their effectiveness. DDDNs are widely used in analyzing the structural and functional characteristics of complex networks in various fields such as biology, social sciences, and computer science. We considered the FN*, M2* , and HMN topological indices for DDDNs. Our computations’ findings provide a clear understanding of the topology of networks that have received limited study. These computed indices exhibit a high level of accuracy when applied to the investigation of QSPRs and QSARs, as they demonstrate the strongest correlation with the acentric factor and entropy.

*e study of topological indices in graph theory is one of the more important topics, as the scienti.c development that occurred in the previous century had an important impact by linking it to many chemical and physical properties such as boiling point and melting point. So, our interest in this paper is to study many of the topological indices “generalized indices’ network” for some graphs that have somewhat strange structure, so it is called the cog-graphs of special graphs “molecular network”, by .nding their polynomials based on vertex − edge degree then deriving them with respect to x, y, and x y, respectively, after substitution x = y = 1 of these special graphs are cog-path, cog-cycle, cog-star, cog-wheel, cog-fan, and cog-hand fan graphs; the importance of some types of these graphs is the fact that some vertices have degree four, which corresponds to the stability of some chemical compounds. *ese topological indices are .rst and second Zagreb, reduced .rst and second Zagreb, hyper Zagreb, forgotten, Albertson, and sigma indices.

A topological index is a real number that relates to a graph that must be a structural invariant. In this paper, we first define a new graph, which is a concept from the coronavirus, called a corona graph. We give some theoretical results for the Wiener and the hyper Wiener index of a graph. Moreover, we calculate some topological indices degree-based on a corona graph. In addition, we are introducing a new topological index 𝑆𝐾4, which is inspired by the definition of the 𝑆𝐾1 index.

One of the more exciting polynomials among the newly presented graph algebraic polynomials is the M−Polynomial, which is a standard method for calculating degree−based topological indices. In this paper, we define the Mve−polynomials based on vertex–edge degree and derive various vertex–edge degree based topological indices from them. Thus, for any graph, we provide some relationships between vertex–edge degree topological indices. Also, we discuss the general Mve−polynomial of r−regular simple graph. Finally, we computed the Mve−polynomial of the 2−ary tree graph

The nullity (degree of singularity) η(G) of a graph G is the multiplicity of zero as an eigenvalue in its spectrum. It is proved that, the nullity of a graph is the number of non-zero independent variables in any of its high zero-sum weightings. Let u and v be nonadjacent coneighbor vertices of a connected graph G, then η(G) = η(G−u) + 1 = η(G−v) + 1. If G is a graph with a pendant vertex (a vertex with degree one), and if H is the subgraph of G obtained by deleting this vertex together with the vertex adjacent to it, then η(G) = η(H). Let H be a graph of order n and G1, G2,…, Gn be given vertex disjoint graphs, then the expanded graph inG H is a graph obtained from the graph H by replacing each vertex vi of H by a graph Gi with extra sets of edges Si,j for each edge vivj of H in which Si,j = {uw: u∈V(Gi), w∈V(Gj)}. In this research, we evaluate the nullity of expanded graphs, for some special ones, such as null graphs, complete bipartite graphs, star graphs, complete graphs, nut graphs, paths, and cycles.