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.