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.

Optimizing regional economic resources is a crucial aspect of the Belt and Road initiative. This paper develops a multi-objective optimization model to objectively evaluate the development level of regional economic resource optimization in Belt and Road countries and to identify the key influencing factors. The model maximizes regional economic and social benefits under constraints of resource availability, output capacity, and coordinated regional development, and it incorporates a synergy measure to ensure robust progress. Our findings show that the regional economic benefits index increased from 0.264 in 2017 to 0.575 in 2023 (a growth rate of 117.8%), while social benefits grew by 14.29%. Additionally, panel regression analysis reveals that merchandise trade, foreign direct investment, road traffic mortality, and industrial development all have significant negative impacts on the optimization of economic resources, at the 1% significance level.

Alzheimer’s disease (AD) is a progressive neurodegenerative condition that affects the elderly population. The early detection and diagnosis of AD is critical for achieving effective treatment, as it can greatly improve the patient experience. AD can be viewed through imaging techniques like MRI, PET, and SPECT, providing valuable information about structural and functional changes. These findings are important in understanding this area. However, each imaging modality offers a different perspective. This information can be better collected from several of the other modalities as well as from some others to improve accuracy and reliability in AD detection. By combining information from different imaging modalities, such as MRI, PET, DTI, and fMRI, automated multimodal medical image frameworks aim to create a fused representation that preserves the relevant features from each modality. Convolutional neural networks (CNNs) and generative adversarial networks (GANs), among other deep learning techniques, have been prevalent in these frameworks for learning discriminative and informative features from multi-modal data. In this paper, The Alzheimer’s Disease Neuroimaging Initiative (ADNI) is used for experimental analysis. The proposed work gives 98.94% of accuracy and 1.06% of error which is greater than the existing approaches.

The power of the public key cryptosystem based on Paley graphs is due to several mathematical problems namely quadratic residuosity, local equivalence, and identification of the graphs induced by a sequence of local complementations of the Paley graphs. The classification in terms of degree of these induced graphs can be useful in the cryptanalysis part of the proposed public-key cryptosystem based on these algebraic graphs. This work aims to give the exact value of the minimum and maximum degree by local complementation, then the possible classifications in terms of degree to the graphs induced by a sequence of local complementations of Paley graphs of degree p less than or equal to 13 and some information about the equivalence problem.

Given a graph \(G \), a set is \(\Delta \) convex if there is no vertex \(u\in V(G)\setminus S \) that forms a triangle with two vertices of \(S \). The \(\Delta \)-convex hull of \(S \) is the minimum \(\Delta \)-convex set containing \(S \). This article is an attempt to discuss the Carath\’eodory number and exchange number on various graph families and standard graph products namely Cartesian, strong and lexicographic products of graphs.

Directed strongly regular graphs were introduced by Duval in 1998 as one of the possible generalization of classical strongly regular graphs to the directed case. Duval also provided several construction methods for directed strongly regular graphs. In this paper, an infinite family of directed strongly regular graphs is constructed, as generalized Cayley graphs.