For any positive integer \(h\), a graph \(G=(V,E)\) is said to be \(h\)-magic if there exists a labeling \(l:E(G)\to \mathbb{Z}_h -\{0\}\) such that the induced vertex set labeling \(\ l^+ : V(G) \to \mathbb{Z}_h \) defined by
\[
l^+ (v)=\sum_{uv \in E(G)} \ l(uv)
\]
is a constant map. The integer-magic spectrum of a graph \(G\), denoted by \(IM(G)\), is the set of all \(h \in \mathbb{N}\) for which \(G\) is \(h\)-magic. So far, only the integer-magic spectra of trees of diameter at most five have been determined. In this paper, we determine the integer-magic spectra of trees of diameter six and higher.

A total Roman \(\{2\}\)-dominating function on a graph \(G = (V,E)\) is a function \(f:V\rightarrow\{0,1,2\}\) with the properties that (i) for every vertex \({v}\in V\) with \(f({v})=0\), \(f(N({v}))\ge2\) and (ii) the set of vertices with \(f({v})>0\) induces a subgraph with no isolated vertices. The weight of a total Roman \(\{2\}\)-dominating function is the value \(f(V)=\sum_{{v}\in V}f({v})\), and the minimum weight of a total Roman \(\{2\}\)-dominating function is called the total Roman \(\{2\}\)-domination number and denoted by \(\gamma_{tR2}(G)\). In this paper, we prove that for every graph \(G\) of order \(n\) with minimum degree at least two, \(\gamma_{tR2}({G})\leq \frac{5n}{6}\).

The penetration of virtual classroom teaching into German teaching is the presentation of teaching innovation in the information age. In this work, we explore the necessity of applying virtual classroom teaching in German classrooms and the effective strategies of German teaching innovation in virtual classrooms, to provide some suggestions for the reform of German teaching. First, the GPS trajectories are transformed into a sequence of hotspot regions using the spatiotemporal properties of GPS points. Then, a sequential pattern mining algorithm of asynchronous cycles with multiple minimum supports based on pattern growth is adopted, and the sequential patterns of asynchronous cycles are deeply recursively mined according to the multiple minimum supports. Experiments show that the proposed IoT-assisted teaching scheme can effectively integrate equipment resources, mine spatiotemporal information, and help students and teachers establish a new educational method of integrating space and land. Compared with the baseline, it can fully exploit the characteristics of German.

The limitations of existing procedures make it difficult to locate and identify old subterranean culverts in urban infrastructure management. In order to effectively manage urban infrastructure, subterranean pipe culverts must be accurately located and detected. In this research, we investigate the method of computing the shortest distance from the point to the ellipse and propose a pipeline collision detection method based on the projection of the direction of the common perpendicular. In the positioning accuracy test, we simulate the detection of straight and curved paths and obtain satisfactory results; the experimental results show that the detection errors are within acceptable limits for different azimuth and bending angles; in the correctness test, we compared with AutoCAD and ArcGIS, and found that the algorithm in this paper shows superiority in collision detection, especially when dealing with complex spatial relationships and large amounts of data, with evident efficiency advantages. Through theoretical analysis and experimental verification, we demonstrate the effectiveness and reliability of the method.

We identified, via a computer search, 143 excluded minors of the spindle surface, the space formed by the identification of two points of the sphere. Per our search, any additional excluded minors must have at least 12 vertices and 28 edges. We also identified 847 topological obstructions for the spindle surface. We conjecture that our lists of excluded minors and topological obstructions are complete.

This paper analyzes the prediction model of enterprise human resource demand based on Internet of Things (IoT) technology and data mining technology. It also analyzes the impact of the company’s growth scale and other key factors on the demand for human resources, tries to establish a coupling factor model of enterprise development and economic benefits, and then analyzes and forecasts the enterprise’s personnel structure and quality structure. The experimental results show that the optimized human resource demand forecasting model integrates the advantages of the grey system model in data processing, can mine the inherent laws of unorganized data information, and provides a certain convenience for forecasting. Through the linear mapping and processing of sample data, the input and output reflect a kind of correlation, thus changing the fault tolerance of information, making the prediction in the calculation process more accurate, and its comprehensive accuracy can exceed 92.5%.

Low efficiency and poor accuracy are caused by missing data in traditional 3D reconstruction methods. This study suggests a new 3D point cloud recognition technique for substation equipment based on 3D laser scanning point clouds, which combines the k-nearest neighbour (KNN) classification algorithm and particle swarm optimisation (PSO) algorithm, to address these issues. The particle swarm optimisation algorithm optimises the coefficient weights of each subspace feature. The k-nearest neighbour classification algorithm is then used to finish the classification. To confirm the superiority and accuracy of the suggested approach, the impact of the point cloud subspace’s size and loss rate on the recognition effect is examined experimentally and contrasted with the enhanced iterative nearest point algorithm. With an average recognition time of 0.19 seconds and a recognition accuracy of over 95\%, the experimental results demonstrate the method’s good performance in terms of efficiency and accuracy, opening up a wide range of potential applications.

Users may receive personalised information services and decision support from personalised recommendations. In this paper, a hybrid algorithm-based personalised recommendation approach for learning English is proposed. The user model is created by merging user interest tags, and the Person Rank algorithm is then recommended based on user information. Second, the question-and-answer model is created once the question-and-answer data has been labelled, and the Problem Rank algorithm is suggested in accordance with the question-and-answer data. Then, the approach of tag-based recommendation, comparable user recommendation, and multi-dimensional sliding window are used to construct the recommendation algorithm model. The experimental findings demonstrate that, following the model’s training with the gradient descent technique, the recommendation accuracy is steady at around 0.78, the suggested information can accommodate users who are learning English, and the personalised recommendation effect is enhanced.

In this paper, we introduce the edge version of doubly resolving set of a graph which is based on the edge distances of the graph. As a main result, we computed the minimum cardinality \(\psi_E\) of edge version of doubly resolving sets of family of \(n\)-sunlet graph \(S_n\) and prism graph \(Y_n\).

Consider the simple connected graph G with vertex set V(G) and edge set E(G). A graph \(G\) can be resolved by \(R\) if each vertex’s representation of distances to the other vertices in \(R\) uniquely identifies it. The minimum cardinality of the set \(R\) is the metric dimension of \(G\). The length of the shortest path between any two vertices, x, y in V(G), is signified by the distance symbol d(x, y). An ordered k-tuple \(r(x/R)=(d(x,z_1),d(x,\ z_2),…,d(x,z_k))\) represents representation of \(x\) with respect to \(R\) for an ordered subset \(R={\{z}_1,z_2,z_3…,z_k\}\) of vertices and vertex \(x\) in a connected graph. Metric dimension is used in a wide range of contexts where connection, distance, and connectedness are essential factors. It facilitates understanding the structure and dynamics of complex networks and problems relating to robotics network design, navigation, optimization, and facility location. Robots can optimize their localization and navigation methods using a small number of reference sites due to the pertinent idea of metric dimension. As a result, many robotic applications, such as collaborative robotics, autonomous navigation, and environment mapping, are more accurate, efficient, and resilient. A claw-free cubic graph (CCG) is one in which no induced subgraph is a claw. CCG proves helpful in various fields, including optimization, network design, and algorithm development. They offer intriguing structural and algorithmic properties. Developing algorithms and results for claw-free graphs frequently has applications in solving of challenging real-world situations. The metric dimension of a couple of claw-free cubic graphs (CCG), a string of diamonds (SOD), and a ring of diamonds (ROD) will be determined in this work.