To address the human activity recognition problem and its application in practical situations, a CNN-LSTM hybrid neural network model capable of automatically extracting sensor data features and memorizing temporal activity data is designed and improved by integrating CNN and gated recurrent units as a variant of RNN. A multi-channel spatiotemporal fusion network-based two-person interaction behavior recognition method is proposed for two-person skeletal sequential behavior recognition. Firstly, a viewpoint invariant feature extraction method is used to extract two-player skeleton features, then a two-layer cascaded spatiotemporal fusion network model is designed, and finally, a multi-channel spatiotemporal fusion network is used to learn multiple sets of two-player skeleton features separately to obtain multi-channel fusion features, and the fusion features are used to recognize the interaction behavior, and the weights are shared among the channels. Applying the algorithm in the paper to the UCF101 dataset for experiments, the accuracy of the two-person cross-object experiment can reach 96.42% and the accuracy of the cross-view experiment can reach 97.46%. The method in the paper shows better performance in two-player interaction behavior recognition compared to typical methods in this field.

Taekwondo behavior recognition has become a popular study issue in the past few decades due to its vast range of applications in the visual realm. The research of Taekwondo behavior recognition based on skeleton sequences has received increasing attention in recent years due to the widespread use of depth sensors and the development of real-time skeleton estimate methods based on depth images. In order to characterize the behavioral sequences, the majority of research work currently in existence extracts the spatial domain information of various skeleton joints within frames and the temporal domain information of the skeleton joints between frames. However, this research work ignores the fact that different joints and postures play different roles in determining the behavioral categories. Consequently, this paper presents a spatio-temporal weighted gesture Taekwondo features-based approach for Taekwondo recognition that employs a bilinear classifier to iteratively compute the weights of the static gestures and joint points relative to the action category in order to identify the joint points and gestures with high information content; concurrently, this paper introduces dynamic temporal regularization and Fourier time pyramid algorithms for temporal modeling in order to provide a better temporal analysis of the behavioural features, and ultimately employs support vector machines to complete the behavioural classification. According to experimental results on several datasets, this strategy outperforms certain other methods in terms of recognition accuracy and is highly competitive.

A large amount of course data has been accumulated in the long-term teaching activities of universities. It is of great research value to use the data resources to analyze the course teaching status and provide decision support for improving the course teaching quality. In this paper, we design and implement a course evaluation system based on association rules and cluster analysis, analyze the functional requirements of the course evaluation system, and pre-process the course evaluation data. Students’ performance data are analyzed by FP-growth association rules, and then clustered by K-means, which can improve the accuracy of data evaluation.The evaluation index system of university English teaching quality under the concept of “Thinking and Government” is established. With the results of the sample survey, the main problems of the evaluation method are summarized and analyzed, and corresponding suggestions are put forward, which provide an important reference for promoting the reform of college English course.

Depression is a clinical disease, mainly accompanied by mood or emotional abnormalities, mainly depression, slow thinking, often accompanied by emotional abnormalities, cognitive behavior, psychophysiological and interpersonal changes or disorders. Here, using static and task-state MRI data, we present a comprehensive study of abnormal neural activity in patients with depression through spatiotemporal, static, and dynamic measures, demonstrating its validity as an underlying biological trait. In order to effectively study the role of emotion regulation in depression, a brain dynamic network synthesis method based on support vector machine model and community detection algorithm was established. We selected data on the mental state of 45 patients from a hospital’s psychiatric disease control center. They had no history of hearing impairment and normal (or corrected) vision. All procedures are agreed in writing by each participant. The results show that this method can effectively reduce the depression degree of the subjects, and the multi-level features of the integration of task activation and task regulation connection reach 81% (\(\mathrm{<}\)0.0010, surrogate test) and 83% (\(\mathrm{<}\)0.0016, surrogate test), respectively. The recovery of its depressive psychological state has a significant impact. Numerous studies have used various forms of emotional stimuli to reveal abnormal behaviors and neural responses in multi-channel emotional processing in patients with depression, providing valuable insights into the mechanism of multi-channel emotion regulation in depression.

In order to provide users with better recommendations, it is particularly important to analyze the behavior of users tagging different resources. In this paper, an attention mechanism based on deep learning is designed to effectively capture the features related to the user’s long-term interests and current interests in the session simultaneously, and alleviate the impact of the user’s interest drift that is difficult to deal with by the current session recommendation algorithm on the recommendation accuracy. The main community discovery algorithms are applied to the clustering analysis of the label system, and their performances on different data sets are compared. Besides, we design a personalized recommendation algorithm for the label system. The experimental results show that the proposed algorithm can find the interests of different users and improve the quality of the recommendation system.

Nowadays, people look at a brand not only to see the value of the brand itself, but to understand the cultural value conveyed behind the brand and experience the connotation of the brand culture. Human-computer interaction technology has also gained more application space with the development of the times. Therefore, a psychological model of brand culture based on human-computer interaction was designed in this paper, and a survey of related content was conducted. In terms of user satisfaction survey, it was concluded that the use of the brand culture mental model based on human-computer interaction technology could greatly improve users’ satisfaction with brand culture and make more people love brand culture; in terms of user participation survey, it was concluded that the brand culture mental model designed based on human-computer interaction technology could achieve better profit results at 21:00 on Sunday. Finally, a survey of user stickiness was carried out, and the test results showed that the brand culture mental model based on human-computer interaction technology established the stickiness between users and brands.

Cities are highly concentrated areas of human civilization, the contradiction between urban development and resources and environment has become increasingly prominent. Inefficient use of energy and land resources, shortage of water resources, and environmental pollution are threatening the healthy development of cities. In this paper, the signal reconstruction algorithm and measurement matrix design in the compressed sensing theory are mainly studied. Aiming at the problems of green city environmental monitoring and landscape design, signal underestimation or overestimation caused by the fixed selection step in the iterative process of sparse adaptive matching tracking algorithm, The threshold idea is introduced into atomic selection, and a variable step size strategy is proposed based on the change of step size. The experimental results show that the establishment of the green city environment monitoring and landscape design model system dynamically changes the network topology, so that data can be transmitted in the mobile ad hoc network.

Deep learning is based on scientific educational psychology theory and is an important concept in contemporary learning theory. Therefore, combining in-depth learning with teaching of political courses, to explore teaching strategies of college political courses based on students’ in-depth learning, requirements for implementing new curriculum standards for cultivating students’ core literacy of disciplines, and cultivating students who meet development requirements of times. High-quality talents are of great value and significance. Through questionnaires and sample interviews, this paper focuses on analyzing specific measures for improvement from four aspects: sufficient teaching preparation, effective teaching implementation, scientific teaching evaluation and normalized teaching reflection. It is highly effective and feasible to increase level of students’ deep learning ability to more than 14.65%.

The development of information society requires the reform of traditional English education. The progress of science and technology, especially the progress of Internet information technology, has penetrated into the English teaching system, reconstructed the relationship between the elements of the English teaching system, and provided technical support for the reform of English teaching. These two aspects are the internal and external impetus of English teaching reform. According to the theory of knowledge construction and multimodal information fusion, this research establishes a user-centered knowledge space, which can respond to user needs quickly, emphasizes the integration and integration of multimodal subject knowledge in resource organization, and expresses multi-dimensional relevance in functional form. The experimental results show that the optimized BOPPPS English teaching model is conducive to improving students’ participation in English classroom interaction. In the new information integration technology environment, students are more likely to put forward opinions or suggestions, and the transformation relationship of students’ interactive behavior becomes more complex.

At present, countries all over the world attach great importance to cultural works. These works have become an important engine of economic development and can make good contributions to economic growth. The traditional tracing control scheme of cultural creation works has some problems, such as incomplete information collection, critical point of unit tracing, information fraud, centralized data storage and so on. At the same time, there seems to be a series of problems that can be solved. This paper analyzes the current situation of the review data of cultural creation products, and puts forward the review and analysis scheme of cultural creation works based on large-scale data algorithm and block chain technology. In addition, by combining department chain technology with intranet and traditional database system, an information database about the supply chain of cultural products in crop departments is established. The processing, logistics and sales information and the information of participants are interrelated.

This paper analyzed the components of the connotation of innovation ability, then constructs a linear spatial model of innovation and entrepreneurship ability, proposes a multi-objective function model of the utilization efficiency and allocation efficiency of education resources, and uses the grey correlation algorithm The experimental simulation and model solution are carried out. The simulation results show that, through the optimization, the utilization efficiency and allocation efficiency of the educational resources for innovation and entrepreneurship for all are increased by 18.72% and 20.98% respectively, and tend to be in equilibrium, which can achieve the optimization of educational resources allocation. Among all people, the correlation value with ideal entrepreneurship is 0.3177, achieving the most excellent innovation and entrepreneurship education.

IMO Member State audits aim to identify non-compliant behavior with the requirements of relevant instruments, enabling the implementation of corrective measures to enhance performance. However, the complexity and diversity of IMO instruments’ requirements result in low evaluation effectiveness and efficiency in current assessment methods of implementation of IMO instruments. To address this challenge, this study proposes a meta-learning model based on prototype networks, focusing on the corrective measures outlined in consolidated audit summary reports approved and issued by the IMO Secretariat. The suggested model conducts meta-learning using small samples, offering a swift and straightforward assessment method. It facilitates the fine classification of corrective measures, providing a way for the consistent and effective assessment of various countries’ current implementation practices. Empirical results of two strategies demonstrate improved classification accuracy. In comparison with traditional manual evaluation, the proposed method achieves accuracy value 71.61% and 65.78% in two strategies respectively. Furthermore, the model exhibits varying prediction accuracy across different articles and demonstrates robust generalization capabilities, highlighting its practicality.

A mapping \(l : E(G) \rightarrow A\), where \(A\) is an abelian group written additively, is called an edge labeling of the graph \(G\). For every positive integer \(h \geqslant 2\), a graph \(G\) is said to be zero-sum \(h\)-magic if there is an edge labeling \(l\) from \(E(G)\) to \(\mathbb{Z}_{h} \backslash \{0\}\) such that \(s(v) = \sum_{uv\in E(G)}l(uv) = 0\) for every vertex \(v \in V(G)\). In 2014, Saieed Akbari, Farhad Rahmati and Sanaz Zare proved that if \(r\) \((r\not= 5)\) is odd and \(G\) is a \(2\)-edge connected \(r\)-regular graph, \(G\) admits a zero-sum 3-magic labeling, and they also conjectured that every 5-regular graph admits a zero-sum \(3\)-magic. In this paper, we first prove that every 5-regular graph with every edge contained in a triangle must have a perfect matching, and then we denote the edge set of the perfect maching by \(EM\), and we make a labeling \(l : E(EM) \rightarrow {2}\), and \(E(E(G) – EM) \rightarrow {1}\). Thus we can easily see this labeling is a zero-sum 3-magic, confirming the above conjecture with a moderate condition.

Let \(\Gamma_{G}\) be the orbit graph of \(G\), with non-central orbits in the subset of order two commuting elements in \(G\), and the vertices of \(\Gamma_{G}\) connected if they are conjugate. The main objective of this study is to compute several topological indices for the orbit graph of a dihedral group, including the Wiener index, the Zagreb index, the Schultz index, and others. We also develop a relationship between the Wiener index and the other indices for the dihedral group’s orbit graph. Furthermore, their polynomial has been computed as well.

\(Y_k\)-tree is defined as \((v_1, v_2,\ldots, v_{k-1};\, v_{k-2} v_k)\) by taking their vertices as \((v_1,\,v_2,\ldots,\,v_k)\) and edges as \(\{(v_1v_2, v_2v_3,\ldots, v_{k-2}v_{k-1})\cup (v_{k-2}v_k)\}\). It is also represented as \((P_ {k-1} +e)\). One can obtain the necessary condition as \(mn(m-1)(n-1)\equiv 0 \pmod {2(k-1)}\), for \(k \geq 5\) to establish a \(Y_k\)-tree decomposition in \(K_m \times K_n\). Here the tensor product is denoted by \(\times\). In this manuscript, it is shown that a \(Y_5\)-tree (gregarious \(Y_5\)-tree) decomposition exists in \(K_m \times K_n\), if and only if, \(mn(m-1)(n-1)\equiv 0 \pmod8\).

For graphs \(F\) and \(H\), the proper Ramsey Number \(PR(F,H)\) is the smallest integer \(n\) so that any \(\chi'(H)\)-edge-coloring on \(K_n\) contains either a monochrome \(F\) or a properly colored \(H\). We determine the proper Ramsey number of \(K_3\) against \(C_3\) and \(C_5\).

We have constructed Block structured Hadamard matrices in which odd number of blocks are used in a row (column). These matrices are different than those introduced by Agaian. Generalised forms of arrays developed by Goethals-Seidel, Wallis-Whiteman and Seberry-Balonin heve been employed. Such types of matrices are applicable in the constructions of nested group divisible designs.

The primary challenge in credit analysis revolves around uncovering the correlation between repayment terms and yield to maturity, constituting the interest rate term structure-an essential model for corporate credit term evaluation. Presently, interest rate term structures are predominantly examined through economic theoretical models and quantitative models. However, predicting treasury bond yields remains a challenging task for both approaches. Leveraging the clustering analysis algorithm theory and the attributes of an insurance company’s customer database, this paper enhances the K-means clustering algorithm, specifically addressing the selection of initial cluster centers in extensive sample environments. Utilizing the robust data fitting and analytical capabilities of the Gaussian process mixture model, the study applies this methodology to model and forecast Treasury yields. Additionally, the research incorporates customer credit data from a property insurance company to investigate the application of clustering algorithms in the analysis of insurance customer credit.

In this paper, we propose a method for effectively evaluating the quality of business English teaching in colleges and universities. The approach is based on a multimodal neural network model integrated with grey correlation analysis. By employing the optimal data clustering criterion, we identify teaching quality evaluation indices. Subsequently, we establish a teaching quality evaluation index system using a genetic algorithm (GA) optimized Radial Basis Function (RBF) neural network. Grey correlation analysis is then applied to assess the quality of business English teaching by considering the relationship between the correlation degree and the evaluation level. The results indicate a correlation degree exceeding 0.90, signifying excellent teaching quality. The reliability of the selected evaluation indicators, assessed through retesting, surpasses 0.700, validating the evaluation results.

A crucial component of kindergarten instruction, collective teaching activities are a good way to educate young children on their overall development. The language field is one of the subjects taught in kindergarten, and it has to do with how kids learn to read, write, and speak. In order to improve teachers’ comprehension of children’s emotional reactions and language, this paper combines quantitative and qualitative methods to observe and analyze the quality of current language collective teaching activities in kindergartens. It also suggests knowledge logic and psychological logic for grasping the content of language collective teaching in kindergartens. To improve the quality of language teaching in kindergartens, it is crucial to adopt a variety of teaching strategies and organizational techniques, provide the proper tools and materials for language learning, pay attention to the key experiences of children learning the language, and enhance learning quality.

In the new era characterized by the modernization of national governance, fair competition is the inherent requirement of building a modern market system. However, the abuse of administrative power by administrative organs to excessively interfere in free-market competition is widespread, seriously damaging the market competition order in China. To avoid the unreasonable intervention of administrative organs in the market economy, restrain the administrative acts of administrative organs, and form a highly “competitive” market environment, the fair competition review system came into being. With the rapid development of blockchain technology, new ideas are provided for the research of fair-trade protocols. Aiming at the system performance bottleneck and high-cost problems caused by centralized processing in traditional fair transaction schemes based on trusted third parties, a fair transaction scheme based on fuzzy signature is proposed. In the proposed scheme, the signature model uses concurrent signature, and both parties hold their own key numbers, which are released through blockchain transactions to bind their signatures. In the whole process, both parties can complete the contract signing without the assistance of a centralized third party. Based on analyzing the security of the proposed scheme, the performance of the proposed scheme is further compared with other similar schemes of the same kind, which shows that the proposed scheme has higher computational efficiency.

Selecting the user comment information of short videos with top 2 likes in the top 50 topics about public cultural services in Shake App as the research object, and facilitating video platforms to identify the high and low quality of the videos and make reasonable promotion arrangements by predicting the short-term playback volume of pop-up videos and analyzing the influencing factors, which is conducive to improving the platform’s pop-up video services and economic benefits. The data related to B station videos are captured, and feature selection and different algorithms are combined to construct random forest model, XG Boost model and LSTM model to predict the playback volume of the pop-up videos, and compare and analyze the effects of different feature combinations on the experimental results. The results show that the prediction accuracy of the random forest model is higher than that of the XG Boost model and the LSTM model, and the features of the pop-up video itself have the greatest influence on the playback volume, while the features of the video markup text have a smaller degree of influence on the playback volume.

The rapid economic progress and widespread use of sophisticated technology elevate the output value per kWh of electricity consumed, underscoring the paramount importance of maintaining an uninterrupted and dependable power supply to avoid substantial economic losses for consumers and society. Investigating the reliability of urban distribution systems emerges not only as a pivotal factor in enhancing power supply quality but also as a cornerstone of electric power modernization, significantly impacting production, technology, and management within the industry while bolstering its economic and social benefits. This study adopts a multifaceted approach: firstly, establishing a methodology for grid-side storage capacity distribution to mitigate substation load factors and implement peak shaving, thereby minimizing load discrepancies. Secondly, it develops a mathematical model encompassing diverse user distributions, employing analytical techniques to derive reliability indices and optimal segment numbers tailored to different user distributions. The research proposes segment optimization based on user distributions, considering both economic viability and reliability, showcasing an interdisciplinary amalgamation of combinatorial principles and scientific computing methodologies. This approach aims to optimize segment distribution, enhancing the reliability and economic feasibility of urban distribution networks through advanced mathematical and computational techniques.

This study introduces a novel approach to address deficiencies in prior teaching quality assessment systems by establishing a mathematical model for evaluation. Utilizing a neural network trained via a particle swarm optimization algorithm (PSO), the method develops a BP (Backpropagation) model fine-tuned by PSO to capture the intricate relationships among diverse indicators influencing teachers’ teaching quality assessment and resulting evaluations. Empirical findings highlight the effectiveness of artificial neural networks in constructing a comprehensive evaluation framework accommodating a wide spectrum of systematic assessments. This approach not only optimizes teaching methodologies but also augments overall teaching efficacy and the quality of educational delivery in a holistic manner. Moreover, it fosters the cultivation of multifaceted individuals proficient in English application skills, contributing to the development of high-quality talent in practical and complex domains. The convergence of advanced mathematical modeling techniques and computational methods, alongside the utilization of numerous indicators, aligns with combinatorial principles, exploring the permutations and relationships of diverse factors impacting teaching quality assessment.

In this paper, we addresses the growing importance of enterprise equipment asset management efficiency. Proposing an advanced approach rooted in combinatorial principles and scientific computing, the study introduces a comprehensive evaluation model for equipment value. Overcoming the limitations of traditional models, a fuzzy algorithm establishes a three-dimensional cross-compound element, encompassing equipment reliability, stability, and accuracy. Hierarchical analysis and the entropy power method determine weights for evaluation indexes, facilitating a quantitative assessment of measurement and production equipment health. Validation through a real energy meter production line demonstrates the model’s effectiveness in comparison to real defect rates. This innovative evaluation model not only offers asset managers a new method for assessing equipment assets but also presents a forward-looking strategy for enterprises to enhance their asset management proficiency, emphasizing the synergies between combinatorics and scientific computing in addressing contemporary economic challenges.

Let \(G = (V, E)\) be a graph with \(n\) vertices. A bijection \(f : V \to \{1, 2, \dots, n\}\) is called a distance magic labeling of \(G\) if there exists an integer \(k\) such that \(\sum_{u \in N(v)}f (u) = k\) for all \(v \in V\), where \(N(v)\) is the set of all vertices adjacent to \(v\). Any graph which admits a distance magic labeling is a distance magic graph. The existence of regular distance magic graphs of even order was solved completely in a paper by Fronček, Kovář, and Kovářová. In two recent papers, the existence of \(4\)-regular and of \((n-3)\)-regular distance magic graphs of odd order was also settled completely. In this paper, we provide a similar classification of all feasible odd orders of \(r\)-regular distance magic graphs when \(r=6,8,10,12\). Even though some nonexistence proofs for small orders are done by brute force enumeration, all the existence proofs are constructive.

A good set on \(k\) vertices is a vertex induced subgraph of the hypercube \(Q_n\) that has the maximum number of edges. The long-lasting problem of characterizing graphs that are cover graphs of lattices is NP-complete. This paper constructs and studies lattice theoretic properties of a class of lattices whose cover graphs are isomorphic to good sets.

Combinatorial mathematics is a versatile field that can provide valuable insights and techniques in various aspects of artificial intelligence and educational research. We focus our attention on the exploration of the mechanism of the role of teachers’ emotional labor In this paper, we merge two parts of data, predicted and formally administered, based on the optimization and management of artificial intelligence English teachers’ emotional labor for the corresponding statistical analysis. Yes individual college English teachers are working for non-interpersonal issues for emotional regulation, temporarily restraining anger and cursing impulses, and communicating with students in a pleasant manner. In the case study of this paper, a teacher repeatedly failed in teaching, but he restrained his frustration and continued to work hard, and finally finished.

In order to determine the optimal scale for urban ride-hailing services and taxis while promoting their sustainable growth, we have developed a Lotka-Volterra evolutionary model that accounts for the competitive, cooperative, and mixed dynamics between these two entities. This model is rooted in the theory of synergistic evolution and is supported by data simulation and analysis. By employing this model, we can identify the appropriate size for urban ride-hailing services and taxis when they reach equilibrium under different environmental conditions. The study’s findings reveal that the evolutionary outcomes of online ride-hailing services and traditional taxis are closely linked to the competitive impact coefficient and the cooperative effect coefficient. In highly competitive environments, intense rivalry can lead to the elimination of the less competitive party, while the dominant player ultimately attains a specific size threshold. As competition moderates, both entities can achieve a balanced and stable coexistence in the market. In cooperative environments, both online ride-hailing services and traditional taxis have more room for development, which facilitates the integration of existing and innovative business models. In environments marked by competition, the development trends of both entities mirror those in competitive settings, but cooperation can slow down the decline of the less competitive party. In conclusion, we propose strategies to foster fair competition between online ride-hailing services and traditional taxis, consider the coexistence of old and new business models, and promote their integrated development.

A vertex labeling \(\xi\) of a graph \(\chi\) is referred to as a ‘vertex equitable labeling (VEq.)’ if the induced edge weights, obtained by summing the labels of the end vertices, satisfy the following condition: the absolute difference in the number of vertices \(v\) and \(u\) with labels \(\xi(v)= a\) and \(\xi(u)= b\) (where \(a,\ b\in Z\)) is approximately \(1\), considering a given set \(A\) that consists of the first \(\lceil \frac{q}{2} \rceil\) non negative integers. A graph \(\chi\) that admits a vertex equitable labeling (VEq.) is termed a ‘vertex equitable’ graph. In this manuscript, we have demonstrated that graphs related to cycles and paths are examples of vertex-equitable graphs.

Network theory is the study of graphs such as representing equilibrium relationships or unequal relationships between different objects. A network can be defined as a graph where nodes and / or margins have attributes (e.g. words). Topological index of a graph is a number that helps to understand its topology and a topological index is known as irregularity index if it is greater than zero and topological index of graph is equal to zero if and only if graph is regular. The irregularity indices are used for computational analysis of nonregular graph topological composition. In this paper, we aim to compute topological invariants of some computer related graph networks. We computed various irregularities indices for the graphs of OTIS swapped network \(OP_a\) and Biswapped Networks \(Bsw(Pa).\)

Let \(G=(V,\,E)\) be a simple graph with vertex set \(V(G)\) and edge set \(E(G)\). The Lanzhou index of a graph \(G\) is defined by \(Lz(G)=\sum\limits_{u \in V(G)} d_u^2\overline{d}_u\), where \(d_u\) (\(\overline{d}_u \) resp.) denotes the degree of the vertex \(u\) in \(G\) (\(\overline{G}\), the complement graph of \(G\) resp.). It has predictive powers to provide insights of chemical relevant properties of chemical graph structures. In this paper we discuss some properties of Lanzhou index. Several inequalities having lower and upper bound for the Lanzhou index in terms of first, second and third Zagreb indices, radius of graph, eccentric connectivity index, Schultz index, inverse sum indeg index and symmetric division deg index, are discussed. At the end the Lanzhou index of corona and join of graphs have been derived.

We define an extremal \((r|\chi)\)-graph as an \(r\)-regular graph with chromatic number \(\chi\) of minimum order. We show that the Turán graphs \(T_{ak,k}\), the antihole graphs and the graphs \(K_k\times K_2\) are extremal in this sense. We also study extremal Cayley \((r|\chi)\)-graphs and we exhibit several \((r|\chi)\)-graph constructions arising from Turán graphs.

A dominating broadcast of a graph \(G\) is a function \(f : V(G) \rightarrow \lbrace 0, 1, 2, \dots ,\text{diam}(G)\rbrace\) such that \(f(v) \leqslant e(v)\) for all \(v \in V(G)\), where \(e(v)\) is the eccentricity of \(v\), and for every vertex \(u \in V(G)\), there exists a vertex \(v\) with \(f(v) > 0\) and \(\text{d}(u,v) \leqslant f(v)\). The cost of \(f\) is \(\sum_{v \in V(G)} f(v)\). The minimum of costs over all the dominating broadcasts of \(G\) is called the broadcast domination number \(\gamma_{b}(G)\) of \(G\). A graph $G$ is said to be radial if \(\gamma_{b}(G)=\text{rad}(G)\). In this article, we give tight upper and lower bounds for the broadcast domination number of the line graph \(L(G)\) of \(G\), in terms of \(\gamma_{b}(G)\), and improve the upper bound of the same for the line graphs of trees. We present a necessary and sufficient condition for radial line graphs of central trees, and exhibit constructions of infinitely many central trees \(T\) for which \(L(T)\) is radial. We give a characterization for radial line graphs of trees, and show that the line graphs of the \(i\)-subdivision graph of \(K_{1,n}\) and a subclass of caterpillars are radial. Also, we show that \(\gamma_{b}(L(C))=\gamma(L(C))\) for any caterpillar \(C\).

In this paper we introduce the concept of independent fixed connected geodetic number and investigate its behaviours on some standard graphs. Lower and upper bounds are found for the above number and we characterize the suitable graphs achieving these bounds. We also define two new parameters connected geo-independent number and upper connected geo-independent number of a graph. Few characterization and realization results are formulated for the new parameters. Finally an open problem is posed.

Let \(E(H)\) and \(V(H)\) denote the edge set and the vertex set of the simple connected graph \(H\), respectively. The mixed metric dimension of the graph \(H\) is the graph invariant, which is the mixture of two important graph parameters, the edge metric dimension and the metric dimension. In this article, we compute the mixed metric dimension for the two families of the plane graphs viz., the Web graph \(\mathbb{W}_{n}\) and the Prism allied graph \(\mathbb{D}_{n}^{t}\). We show that the mixed metric dimension is non-constant unbounded for these two families of the plane graph. Moreover, for the Web graph \(\mathbb{W}_{n}\) and the Prism allied graph \(\mathbb{D}_{n}^{t}\), we unveil that the mixed metric basis set \(M_{G}^{m}\) is independent.

Consider a total labeling \(\xi\) of a graph \(G\). For every two different edges \(e\) and \(f\) of \(G\), let \(wt(e) \neq wt(f)\) where weight of \(e = xy\) is defined as \(wt(e)=|\xi(e) – \xi(x) – \xi(y)|\). Then \(\xi\) is called edge irregular total absolute difference \(k\)-labeling of \(G\). Let \(k\) be the minimum integer for which there is a graph \(G\) with edge irregular total absolute difference labeling. This \(k\) is called the total absolute difference edge irregularity strength of the graph \(G\), denoted \(tades(G)\). We compute \(tades\) of \(SC_{n}\), disjoint union of grid and zigzag graph.

A total dominator coloring of \(G\) without isolated vertex is a proper coloring of the vertices of \(G\) in which each vertex of \(G\) is adjacent to every vertex of some color class. The total dominator chromatic number \(\chi^t_d(G)\) of \(G\) is the minimum number of colors among all total dominator coloring of \(G\). In this paper, we will give the polynomial time algorithms to computing the total dominator coloring number for \(P_4\)-reducible and \(P_4\)-tidy graphs.

An \(H\)-(a,d)-antimagic labeling in a \(H\)-decomposable graph \(G\) is a bijection \(f: V(G)\cup E(G)\rightarrow {\{1,2,…,p+q\}}\) such that \(\sum f(H_1),\sum f(H_2),\cdots, \sum f(H_h)\) forms an arithmetic progression with difference \(d\) and first element \(a\). \(f\) is said to be \(H\)-\(V\)-super-\((a,d)\)-antimagic if \(f(V(G))={\{1,2,…,p\}}\). Suppose that \(V(G)=U(G) \cup W(G)\) with \(|U(G)|=m\) and \(|W(G)|=n\). Then \(f\) is said to be \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic labeling if \(f(U(G))={\{1,2,…,m\}}\) and \(f(W(G))={\{m+1,m+2,…,(m+n=p)\}}\). A graph that admits a \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic labeling is called a \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic decomposable graph. In this paper, we prove that complete bipartite graphs \(K_{m,n}\) are \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic decomposable with both \(m\) and \(n\) are even.

A Grundy \(k\)-coloring of a graph \(G\) is a proper \(k\)-coloring of vertices in \(G\) using colors \(\{1, 2, \cdots, k\}\) such that for any two colors \(x\) and \(y\), \(x<y\), any vertex colored \(y\) is adjacent to some vertex colored \(x\). The First-Fit or Grundy chromatic number (or simply Grundy number) of a graph \(G\), denoted by \(\Gamma \left(G\right)\), is the largest integer \(k\), such that there exists a Grundy \(k\)-coloring for \(G\). It can be easily seen that \(\Gamma \left(G\right)\) equals to the maximum number of colors used by the greedy (or First-Fit) coloring of \(G\). In this paper, we obtain the Grundy chromatic number of Cartesian Product of path graph, complete graph, cycle graph, complete graph, wheel graph and star graph.

Determining the Tutte polynomial \(T(G;x,y)\) of a graph network \(G\) is a challenging problem for mathematicians, physicians, and statisticians. This paper investigates a self-similar network model \(M(t)\) and derives its Tutte polynomial. In addition, we evaluate exact explicit formulas for the number of acyclic orientations and spanning trees of it as applications of the Tutte polynomial. Finally, we use the derived \(T(M(t);x,y)\) to obtain the Tutte polynomial of another self-similar model \(N(t)\) presented in [1] and correct the main result discussed in [1] by Ma et al. and test our result numerically by using Matlab.

A vertex-colouring of a graph \(\Gamma\) is rainbow vertex connected if every pair of vertices \((u,v)\) in \(\Gamma\) there is a \(u-v\) path whose internal vertices have different colours. The rainbow vertex connection number of a graph \(\Gamma\), is the minimum number of colours needed to make \(\Gamma\) rainbow vertex connected, denoted by \(rvc(\Gamma)\). Here, we study the rainbow vertex connection numbers of middle and total graphs. A total-colouring of a graph \(\Gamma\) is total rainbow connected if every pair of vertices \((u,v)\) in \(\Gamma\) there is a \(u-v\) path whose edges and internal vertices have different colours. The total rainbow connection number of \(\Gamma\), is the minimum number of colours required to colour the edges and vertices of \(\Gamma\) in order to make \(\Gamma\) total rainbow connected, denoted by \(trc(\Gamma)\). In this paper, we also research the total rainbow connection numbers of middle and total graphs.

The harmonic index \(H(G)\) of a graph \(G\) is defined as the sum of the weights \(\frac{2}{d_{u}+ d_{v}}\) of all edges \(uv\) of \(G\), where \(d_{u}\) denotes the degree of a vertex \(u\). Delorme et al. [1] (2002) put forward a conjecture concerning the minimum Randić index among all connected graphs with \(n\) vertices and the minimum degree at least \(k\). Motivated by this paper, a conjecture related to the minimum harmonic index among all connected graphs with \(n\) vertices and the minimum degree at least \(k\) was posed in [2]. In this work, we show that the conjecture is true for a connected graph on $n$ vertices with \(k\) vertices of degree \(n-1\), and it is also true for a \(k\)-tree. Moreover, we give a shorter proof of Liu’s result [3].

Let \(L\) be a unital ring with characteristic different from \(2\) and \(\mathcal{O}(L)\) be an algebra of Octonion over \(L\). In the present article, our attempt is to present the characterization as well as the matrix representation of some variants of derivations on \(\mathcal{O}(L)\). The matrix representation of Lie derivation of \(\mathcal{O}(L)\) and its decomposition in terms of Lie derivation and Jordan derivation of \(L\) and inner derivation of \(\mathcal{O}\) is presented. The result about the decomposition of Lie centralizer of \(\mathcal{O}\) in terms of Lie centralizer and Jordan centralizer of \(L\) is given. Moreover, the matrix representation of generalized Lie derivation (also known as \(D\)-Lie derivation) of \(\mathcal{O}(L)\) is computed.

A sum divisor cordial labeling of a graph \(G\) with vertex set \(V(G)\) is a bijection \(f\) from \(V(G)\) to \(\{1,2,\cdots,|V(G)|\}\) such that an edge \(uv\) is assigned the label \(1\) if \(2\) divides \(f(u)+f(v)\) and \(0\) otherwise; and the number of edges labeled with \(1\) and the number of edges labeled with \(0\) differ by at most \(1\). A graph with a sum divisor cordial labeling is called a sum divisor cordial graph. In this paper, we discuss the sum divisor cordial labeling of transformed tree related graphs.

For a graph \(G\) and a positive integer \(k\), a royal \(k\)-edge coloring of \(G\) is an assignment of nonempty subsets of the set \(\{1, 2, \ldots, k\}\) to the edges of \(G\) that gives rise to a proper vertex coloring in which the color assigned to each vertex \(v\) is the union of the sets of colors of the edges incident with \(v\). If the resulting vertex coloring is vertex-distinguishing, then the edge coloring is a strong royal \(k\) coloring. The minimum positive integer \(k\) for which a graph has a strong royal \(k\)-coloring is the strong royal index of the graph. The primary emphasis here is on strong royal colorings of trees.

The coloring of all the edges of a graph \(G\) with the minimum number of colors, such that the adjacent edges are allotted a different color is known as the proper edge coloring. It is said to be equitable, if the number of edges in any two color classes differ by atmost one. In this paper, we obtain the equitable edge coloring of splitting graph of \(W_n\), \(DW_n\) and \(G_n\) by determining its edge chromatic number.

Let us consider a~simple connected undirected graph \(G=(V,E)\). For a~graph \(G\) we define a~\(k\)-labeling \(\phi: V(G)\to \{1,2, \dots, k\}\) to be a~distance irregular vertex \(k\)-labeling of the graph \(G\) if for every two different vertices \(u\) and \(v\) of \(G\), one has \(wt(u) \ne wt(v),\) where the weight of a~vertex \(u\) in the labeling \(\phi\) is \(wt(u)=\sum\limits_{v\in N(u)}\phi(v),\) where \(N(u)\) is the set of neighbors of \(u\). The minimum \(k\) for which the graph \(G\) has a~distance irregular vertex \(k\)-labeling is known as distance irregularity strength of \(G,\) it is denoted as \(dis(G)\). In this paper, we determine the exact value of the distance irregularity strength of corona product of cycle and path with complete graph of order \(1,\) friendship graph, Jahangir graph and helm graph. For future research, we suggest some open problems for researchers of the same domain of study.

Elimination ideals are monomial ideals associated to simple graphs, not necessarily square–free, was introduced by Anwar and Khalid. These ideals are Borel type. In this paper, we obtain sharp combinatorial upper bounds of the Castelnuovo–Mumford regularity of elimination ideals corresponding to certain family of graphs.

Let \(G\) be a simple connected graph with vertex set \(V\) and diameter \(d\). An injective function \(c: V\rightarrow \{1,2,3,\ldots\}\) is called a radio labeling of \(G\) if \({|c(x) c(y)|+d(x,y)\geq d+1}\) for all distinct \(x,y\in V\), where \(d(x,y)\) is the distance between vertices \(x\) and \(y\). The largest number in the range of \(c\) is called the span of the labeling \(c\). The radio number of \(G\) is the minimum span taken over all radio labelings of \(G\). For a fixed vertex \(z\) of \(G\), the sequence \((l_1,l_2,\ldots,l_r)\) is called the level tuple of \(G\), where \(l_i\) is the number of vertices whose distance from \(z\) is \(i\). Let\(J^k(l_1,l_2,\ldots,l_r)\) be the wedge sum (i.e. one vertex union) of \(k\geq2\) graphs having same level tuple \((l_1,l_2,\ldots,l_r)\). Let \(J(\frac{l_1}{l’_1},\frac{l_2}{l’_2},\ldots,\frac{l_r} {l’_r})\) be the wedge sum of two graphs of same order, having level tuples  \((l_1,l_2,\ldots,l_r)\) and \((l’_1,l’_2,\ldots,l’_r)\). In this paper, we compute the radio number for some sub-families of \(J^k(l_1,l_2,\ldots,l_r)\) and \(J(\frac{l_1}{l’_1},\frac{l_2}{l’_2},\ldots,\frac{l_r}{l’_r})\).