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.

Fuzzy graph models enjoy the ubiquity of being in natural and human-made structures, namely dynamic process in physical, biological and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems which are often uncertain, it is highly difficult for an expert to model those problems based on a fuzzy graph (FG). Vague graph structure (VGS) can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. Likewise, VGSs are very useful tools for the study of different domains of computer science such as networking, capturing the image, clustering, and also other issues like bioscience, medical science, and traffic plan. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGSs. Operations are conveniently used in many combinatorial applications. In various situations, they present a suitable construction means; therefore, in this research, three new operations on VGSs, namely, maximal product, rejection, residue product were presented, and some results concerning their degrees and total degrees were introduced. Irregularity definitions have been of high significance in the network heterogeneity study, which have implications in networks found across biology, ecology and economy; so special concepts of irregular VGSs with several key properties were explained. Today one of the most important applications of decision making is in medical science for diagnosing the patient’s disease. Hence, we recommend an application of VGS in medical diagnosis.

The discharging method is most well-known for its central role in the proof of the Four Color Theorem. This proof technique was extensively applied to study various graph coloring problems, in particular on planar graphs. In this paper, we show that suitably altered discharging technique can also be used on domination type problems. The general discharging approach for domination type problems is illustrated on a specific domination type problem, the double Roman domination on some generalized Petersen graphs. By applying this approach, we first prove that γ dR(G) ≥ 3n/Δ(G)+1 for any connected graph G with n ≥ 2 vertices. As examples we also determine the exact values of the double Roman domination numbers of the generalized Petersen graphs P(n,1) and the double generalized Petersen graphs DP(n,1). The obtained results imply that P(n,1) is double Roman if and only if n 6≢ 2 (mod 4) and DP(n,1) is double Roman if and only if n ≡ 0 (mod 4).

Currently, machine learning techniques are widely used in structural seismic response studies. The developed network models for various types of seismic response provide new ways to analyse seismic hazards. However, it is not easy to balance the applicability of the input, accuracy, and computational efficiency for existing network models. In this paper, a neural network model containing an efficient self-adaptive feature extraction module (AFEM) is designed. It can recognize time-frequency features from ground motion (GM) inputs for structural seismic response prediction tasks while considering the model’s computational accuracy and computational cost. The self-adaptive feature extraction module is constructed based on the MFCCs feature extraction process in NLP. AFEM recognizes time-frequency features closely related to structures’ behaviour and response under dynamic loads. Taking the seismic response prediction of a typical building as the target task, the neural network configuration, including a baseline model $M_0$ and three comparison models ($M_1$, $M_2$, and $M_3$) with AFEM, is systematically analysed. The results demonstrate that the proposed $M_1$ model with initial AFEM, the $M_2$ model with combined amplitude and phase features, and the $M_3$ model with a complex-valued network are more adaptable than the baseline model to the target task. The extracted amplitude and phase features by the $M_3$ model’s AFEM significantly improve model validation accuracy by 8.6% while reducing computation time by 11.4%. It could provide the basis for future research on regional earthquake damage intelligence assessment systems.