A nanostructure belong to a significant and an extensively investigated compounds in chemical science. It has been derived through engineering mechanism at the molecular scale. The most important of these new materials are carbon nanotubes. Carbon nanosheets are 2-dimensional lattices of carbon nanotubes. To compute and study topological indices of nanostructures are a respected problem in nanotechnology. Quantitative structure-property and structure-activity relationships of the carbon nanotubes compounds necessitate expressions for the molecular topological features of these compounds. Topological indices are vital devices for investigating chemical compounds to comprehend the fundamental topology of chemical structures. Ev-degree and ve-degree based topological indices are two novel degrees based indices as of late defined in chemical graph theory. In this paper, we have computed topological indices based on ev-degree and ve-degree for the two significant classes of carbon nanosheets composed by C₄ and C₈ denoted by $T^1{C}_4{C}_8(S)[p,q]$ and $T^2{C}_4{C}_8(R)[p,q].$

Topological indices are graph invariants computed by the distance or degree of vertices of the molecular graph. In chemical graph theory, topological indices have been successfully used in describing the structures and predicting certain physicochemical properties of chemical compounds. In this paper, we propose a definition of generalized bridge molecular graphs that can model more kinds of long chain polymerization products than the bridge molecular graphs, and provide some results of the edge versions of atom-bond connectivity ( $AB{C}_e$ ) and geometric arithmetic ( $G{A}_e$ ) indices for some generalized bridge molecular graphs, which have regular, periodic and symmetrical structures. The results of this paper offer promising prospects in the applications for chemical and material engineering, especially in chemical industry research.

The Cartesian product and join are two classical operations in graphs. Let $d_{L(G)}(e)$ be the degree of a vertex $e$ in line graph $L(G)$ of a graph $G$. The edge versions of atom-bond connectivity ($AB{C}_e$) and geometric arithmetic ($G{A}_e$) indices of G are defined as ${\displaystyle \sum _{ef\in E(L(G))}\sqrt{\frac{d_{L(G)}(e)+d_{L(G)}(f)-2}{d_{L(G)}(e)\times d_{L(G)}(f)}}}$ and ${\displaystyle \sum _{ef\in E(L(G))}\frac{2\sqrt{d_{L(G)}(e)\times d_{L(G)}(f)}}{d_{L(G)}(e)+d_{L(G)}(f)}}$, respectively. In this paper, $AB{C}_e$ and $G{A}_e$ indices for certain Cartesian product graphs (such as $P_n\square P_m$, $P_n\square C_m$ and $P_n\square S_m$) are obtained. In addition, $AB{C}_e$ and $G{A}_e$ indices of certain join graphs (such as $C_m+P_n+S_r$, $P_m+P_n+P_r$, $C_m+C_n+C_r$ and $S_m+S_n+S_r$) are deduced. Our results enrich and revise some known results.

Mathematical modeling with the help of numerical coding of graphs has been used in the different fields of science, especially in chemistry for the studies of the molecular structures. It also plays a vital role in the study of the quantitative structure activities relationship (QSAR) and quantitative structure properties relationship (QSPR) models. Todeshine et al. (2010) and Eliasi et al. (2012) defined two different versions of the 1st multiplicative Zagreb index as ∏Γ=∏p∈VΓdΓp2 and ∏1Γ=∏pq∈EΓdΓp+dΓq, respectively. In the same paper of Todeshine, they also defined the 2nd multiplicative Zagreb index as ∏2Γ=∏pq∈EΓdΓp×dΓq. Recently, Liu et al. [IEEE Access; 7(2019); 105479–-105488] defined the generalized subdivision-related operations of graphs and obtained the generalized F-sum graphs using these operations. They also computed the first and second Zagreb indices of the newly defined generalized F-sum graphs. In this paper, we extend this study and compute the upper bonds of the first multiplicative Zagreb and second multiplicative Zagreb indices of the generalized F-sum graphs. At the end, some particular results as applications of the obtained results for alkane are also included.

This paper introduces a method to detect multiple power quality (PQ) disturbances of power system features based on an independent component analysis (ICA) and a sparse auto encoder (SAE). Seven basic single PQ disturbances are extracted as typical features from numerous training samples of PQ disturbances by the SAE method, which can automatically gain the training features rather than manually selecting features as in the conventional approaches. An ICA is adapted to conduct basic separate signals from the blind original disturbance sources. The experimental results indicate that the proposed method has not only achieved a significant improvement for detecting single PQ disturbances but is also effective for detecting and classifying multiple PQ disturbances.