Graph invariants, often regarded as topological indices, play a pivotal role in understanding and quantifying the structural properties of graphs. Among these, the line completion number has emerged as a significant measure of a graph’s edge connectivity and topology. In 1992, Bagga et al. defined a generalization of line graphs, namely super line graphs, and introduced the concept of the line completion number as a topological index of a graph. They calculated the line completion number for several classes of graphs, showcasing its utility in understanding graph structure. The line completion number of a graph, is the smallest index such that the super line graph becomes a complete graph. This index encapsulates the interplay between edge relationships and structural complexity, making it a versatile tool for characterizing graphs. Building upon this foundation, we analogously introduce the concepts of super point graphs and the point completion number, as vertex-centric topological indices. We establish a relationship between the point completion number and the line completion number, further extending the framework of graph invariants. Additionally, we compute the point completion numbers for various graph classes and analyze their structural implications. Our findings emphasize the significance of completion numbers as robust descriptors for graph topology, with potential applications in network analysis, chemistry, and other domains.

In IoT-managed power systems, equipment or communication failures can result in missing or abnormal power quality data, making data restoration increasingly important. Traditional repair methods often struggle to capture complex data relationships and suffer from low accuracy. This paper proposes a power quality data restoration approach based on a low-rank matrix completion algorithm to enhance repair accuracy and efficiency. The system consists of three main steps: data preprocessing, matrix completion, and result validation. Z-score normalization is applied to raw data, and Singular Value Decomposition (SVD) is used for low-rank approximation in matrix filling. Cross-validation and error metrics are employed to assess performance. Experimental results show that at a 10% missing rate, the mean square error is approximately 0.1. The proposed method demonstrates superior performance over traditional approaches, particularly at low missing rates, offering reliable support for monitoring and control in power IoT systems.

The current changes in China’s population structure and dynamics have led to profound challenges in population planning, forecasting, decision-making, and early warning. To address the issues of predicting age- and gender-specific population retention, migration, and birth rates, a combination model of Multilayer Perceptron (MLP) and Random Forest (RF) is constructed using stacking techniques, with a discrete population development equation as the base model. The MLP-RF model is employed to perform regression training on population data, resulting in a novel ensemble approach to population forecasting. The study uses the data from the sixth and seventh national censuses of Hebei Province, reconstructing population data for 2010-2020. After data training and error evaluation, it is demonstrated that the ensemble forecasting model has excellent predictive capabilities for population retention, migration, and birth-related issues.

In the current energy-constrained era, promoting electric vehicles (EVs) is a necessary trend. However, the simultaneous and uncoordinated charging of diverse EVs can negatively impact the power grid. This paper proposes a scaled EV orderly scheduling model, comprising charging demand simulation and a scheduling algorithm. Monte Carlo simulation, based on charging probability models, is used to generate EV cluster entry information and preprocess parameters. Two control strategies are proposed for clean energy dispatch and EV-based grid operation, accounting for user behavior-induced load variations. A microgrid optimization model is developed, with economic cost weights calculated. The model is solved using an improved PSO algorithm (APSO). Results show the APSO achieves better performance, with hourly average exchange loads of 2.7092 P/kW (vs. 1.9979 P/kW for PSO). Under 30–80% user responsiveness, microgrid management and environmental costs are reduced to 28,618.439 yuan and 7,864.685 yuan, respectively.

This paper investigates human-computer communication within the framework of deep learning and identifies three key features of such interaction. A cross-cultural empathy feature aliasing model based on Graph Neural Network-Attention Mechanism-Bi-directional Gating Unit (GCN-Attention-BiGRU) is proposed, with categorical cross-entropy and L2 regularization as the loss function. By integrating IoT and deep learning, an adaptive interaction model is developed and evaluated through experiments. Results show high mean scores for empathy (4.537), relevance (4.447), and fluency (4.499) across 60 samples, indicating effective empathy feature extraction. Additionally, the proposed model demonstrates greater efficiency and adaptability compared to traditional interaction models, enhancing cross-cultural empathy in human-computer communication.

In a society governed by the rule of law, constitutional interpretation forms the foundation of judicial practice. This paper focuses on the role of constitutional hermeneutics in shaping judicial practice in China. Using data from 2010 to 2020, an evaluation index system and fuzzy comprehensive evaluation method are employed to assess the development quality of China’s judicial practice. A multi-period DID regression model further examines the impact of constitutional hermeneutics. Results show that development scores ranged from 86.04 to 92.22, reflecting steady improvement in fairness, efficiency, and effectiveness. Constitutional hermeneutics significantly enhanced judicial practice (P < 0.01), with the positive effects of value supplementation and loophole filling confirmed through robustness tests.

Given two graphs \( G_1 \) and \( G_2 \), the size Ramsey number \( \hat{r}(G_1, G_2) \) refers to the smallest number of edges in a graph \( G \) such that for any red-blue edge-coloring of \( G \), either a red subgraph \( G_1 \) or a blue subgraph \( G_2 \) is present in \( G \). If we further restrict the host graph \( G \) to be connected, we obtain the connected size Ramsey number, denoted as \( \hat{r}_c(G_1, G_2) \). Erd\H{o}s and Faudree (1984) proved that \( \hat{r}(nK_2, K_{1,m}) = mn \) for all positive integers \( m, n \). In this paper, we concentrate on the connected analog of this result. Rahadjeng, Baskoro, and Assiyatun (2016) provided the exact values of \( \hat{r}_c(nK_2, K_{1,m}) \) for \( n = 2, 3 \). We establish a more general result: for all positive integers \( m \) and \( n \) with \( m \ge \frac{n^2 + 2pn + n – 3}{2} \), we have \( \hat{r}_c(nK_{1,p}, K_{1,m}) = n(m + p) – 1 \). As a corollary, \( \hat{r}_c(nK_2, K_{1,m}) = nm + n – 1 \) for \( m \ge \frac{n^2 + 3n – 3}{2} \). We also propose a conjecture for the interested reader.

A subset \(S \subset V(G)\) is called a captive dominating set of a graph \(G\) if \(S\) is a total dominating set and every vertex \(v \in S \) is adjacent to at least one vertex which is not in \(S\). Furthermore, a captive dominating set \(S\) is termed a minimal captive dominating set if no proper subset \( S’ \subset S \) qualifies as a captive dominating set. The minimum size of such captive dominating set in \(G\) is referred to as the captive domination number of \(G\), denoted by \( \gamma_{ca}(G)\). This paper investigates the relationship between the captive domination number and the order of a graph. We establish bounds on the captive domination number and present results for specific graph families obtained through various graph operations.

Let\(G\) be an undirected graph. A tree partition of\(G\) is a set of trees whose edge sets are disjoint and whose union is the edge set of\(G\). The minimum cardinality of such a tree partition is called the tree partition number of\(G\). We show that for various types of trees allowed in the tree partition, that the only linear operators that preserve the tree partition number are vertex permutations.

The Mostar index \( \text{MoI} \) of a finite and connected graph \( G \) is a measure of asymmetry, focusing on the edge-based structure of the graph. For an edge \( xy \) in \( G \), let \( \gamma_{xy} \) and \( \gamma_{yx} \) denote the cardinalities of the sets of vertices closer to \( x \) and \( y \) respectively, then the Mostar index is defined as: \( \text{MoI}(G) = \sum_{xy \in E(G)} |\gamma_{xy} – \gamma_{yx}| \) where the summation is taken over all edges \( xy \in G \). This edge-wise difference reflects how asymmetrically the graph is structured around each edge and summing these differences across all edges yields the Mostar index for the graph. In this article, we compute the \( \text{MoI} \) for certain classes of bicyclic graphs that are of particular interest due to their moderately complex structure, lying between acyclic and polycyclic graphs. We classify bicyclic graphs into three distinct types, namely \( \mathcal{B}^{1}(m,n),\; \mathcal{B}^{2}(l,m,n) \) and \( \mathcal{B}^{1}(l,m) \), based on their cycle arrangements and then provide explicit formulas for calculating the exact value of the Mostar index.