Nanoparticles have potential applications in a wide range of fields, including electronics, medicine and material research, because of their remarkable and exceptional attributes. Carbon nanocones are planar carbon networks with mostly hexagonal faces and a few non-hexagonal faces (mostly pentagons) in the core. Two types of nanocone configurations are possible: symmetric and asymmetric, depending on where the pentagons are positioned within the structure. In addition to being a good substitute for carbon nanotubes, carbon nanocones have made an identity for themselves in a number of fields, including biosensing, electrochemical sensing, biofuel cells, supercapacitors, gas storage devices, and biomedical applications. Their astonishing chemical and physical attributes have made them well-known and widely accepted in the fields of condensed matter physics, chemistry, material science, and nanotechnology. Mathematical and chemical breakthroughs were made possible by the concept of modeling a chemical structure as a chemical graph and quantitatively analyzing the related graph using molecular descriptors. Molecular descriptors are useful in many areas of chemistry, biology, computer science, and other sciences because they allow for the analysis of chemical structures without the need for experiments. In this work, the quotient graph approach is used to establish the distance based descriptors of symmetrically configured two-pentagonal and three-pentagonal carbon nanocones.

A kite \(K\) is a graph which can be obtained by joining an edge to any vertex of \(K_3\). A kite with edge set \(\{ab,\,bc,\,ca,\,cd\}\) can be denoted as \((a,\,b,\,c;\,cd)\). If every vertex of a kite in the decomposition lies in different partite sets, then we say that a kite decomposition of a multipartite graph is a gregarious kite decomposition. In this manuscript, it is shown that there exists a decomposition of \((K_m \otimes \overline{K}_n) \times (K_r \otimes \overline{K}_s) \) into gregarious kites if and only if
\[
n^2 s^2 m(m-1)r(r-1) \equiv 0 \pmod{8},
\]
where \(\otimes\) and \(\times\) denote the wreath product and tensor product of graphs respectively. We denote a gregarious kite decomposition as \(\it GK\)-decomposition.

With the rapid development of my country’s socialist market economy, the system of joint and several liability has been established in my country’s civil and commercial law and is playing an increasingly important role. There are also problems such as scattered regulations and contradictory laws and regulations at the level. Since there is no unified application principle established in judicial practice, the litigation burden caused by the recovery lawsuit also wastes a lot of trial resources. Dimensional key features distinguish confusing charges. Use regular expression technology to extract key content such as fact descriptions, defendants’ charges, relevant laws and regulations in legal documents and create JSON format documents; use stammer word segmentation and stop word list to remove stop words; use Word2Vec algorithm to represent text into vector form , establish a judicial judgment prediction model and an optimization model, and through experimental comparison, it is concluded that the performance of the model after adding focal loss is improved by 1.82%, 0.45%, 1.62%, and 1.62% compared with the cross entropy loss, and the final accuracy of the optimized model is 84.78%. , the precision rate is 87%, the recall rate is 85%, and the F1 value is 85%. The system is expected to assist judicial workers in classifying crimes with joint liability and reduce the burden of judicial workers reading many legal documents to classify crimes.

The evolution of computer science and the innovations in language teaching methodologies have paved the way for computer-assisted language learning (CALL) technology to tackle pertinent challenges. While existing CALL systems primarily emphasize vocabulary and grammar acquisition, their evaluation mechanisms often rely on a limited set of criteria, resulting in a simplistic assessment of learners’ pronunciation skills. This oversight underscores the need for a more comprehensive approach. In response, this study targets Chinese college students’ English oral proficiency and aims to enhance the conventional computerized evaluation method. Our approach involves integrating multiple assessment parameters, including pitch, speed, rhythm, and intonation. For instance, pitch assessment is grounded on frequency central feature parameters, while speech speed evaluation considers speech duration, thus enriching the evaluation framework. Through experimental validation, the efficacy of our method in evaluating pitch, speed, rhythm, and intonation has been substantiated, reaffirming its reliability.

The common bills in life include VAT invoices, taxi invoices, train invoices, plane itineraries, machine-printed invoices, etc. Most of these common bills are presented in the form of fixed form templates, so template matching can be used. , for a certain fixed template bill, manually set the rules to determine the spatial position of the key area, extract the corresponding text information, or build a model with logical semantic relationship and spatial relative relationship between the bill texts of different attributes, from the global image of the image. Identify the required key text information in the text information. However, these methods are either limited by fixed ticket templates, or cannot guarantee considerable accuracy. The electronicization of paper invoices mainly needs to go through the steps of text detection, bill recognition and text recognition. Based on this, this paper adopts the DL method. Construct a financial bill recognition model and combine experiments to explore the effectiveness and superiority of the model. The results show that our model can achieve a recognition accuracy rate of up to 91\%, and also achieve a 26\% improvement in recognition speed.

The maximum-weight perfect matching inverse issue in graph theory and text clustering are the two primary topics of this study. We suggest a novel approach to text clustering that makes use of self-encoders and BERT embeddings for feature extraction in order to increase clustering accuracy. According to experimental results, our technique enhances the clustering results greatly and performs well on numerous short text datasets. In the context of graph theory, we examine the unit paradigm inverse issue of maximum-weight perfect matching with value constraints and provide a robust polynomial-time method for its solution. In addition to effectively solving the maximum-weight perfect matching inverse issue, our technique can also produce the best weight vector configuration scheme for real-world uses. In conclusion, our work has advanced the domains of text clustering and graph theory significantly, offering fresh approaches and theoretical underpinnings for future investigations.

A brief survey on mutually orthogonal resolutions of some combinatorial designs is presented. Some \((2,w)\)-threshold schemes from mutually orthogonal resolutions of these designs are also obtained.

In the era of social media platforms like Douyin, preserving the essence of traditional Chinese culture while adapting it to contemporary trends is crucial for its continued relevance and vitality. This paper delves into the practical implications of leveraging social media for cultural communication, emphasizing communication strategies tailored to platforms like Douyin. It introduces two novel algorithms for generating Douyin information dissemination trees: one based on retweeting relationships and another optimized for rapid dissemination. Comparative experiments assess the performance of these algorithms and analyze the node distribution within dissemination trees, aiming to enhance the dissemination power of traditional culture and foster its inheritance and innovation.

The “Three Rural Issues” has always been the top priority of my country’s economic development, which related to the construction process of modern agriculture, the development effect of agricultural economy and the development speed of the national economy. In recent years, the state and local governments have taken the construction of new countryside as the starting point, seriously discussed many problems faced by the agricultural economy in the process of development, and took targeted measures to effectively solve them, which better promoted the construction of new countryside. Agriculture plays an important role in the national economy and is the foundation of national economic development. Under the background of new rural construction, we must strengthen the management of agricultural economy. This paper analyzes the main contents, characteristics and existing problems of agricultural sustainable development under the background of new rural construction, and puts forward solutions that hope to be discussed by a wide range of partners.

Zhou, Xu and Sun [S. Zhou, Y. Xu, Z. Sun, Degree conditions for fractional \((a,b,k)\)-critical covered graphs, Information Processing Letters 152(2019)105838] defined the concept of a fractional \((a,b,k)\)-critical covered graph, namely, a graph \(G\) is a fractional \((a,b,k)\)-critical covered graph if after removing any \(k\) vertices of \(G\), the remaining graph of \(G\) is a fractional \([a,b]\)-covered graph. In this paper, we prove that a graph \(G\) with \(\delta(G)\geq2+k\) is fractional \((2,b,k)\)-critical covered if \(bind(G)>\frac{b+k}{b-1}\), where \(k\geq0\) and \(b\geq2+k\) are two integers.

In this paper, we study the submodular hitting set problem (SHSP), which is a variant of the hitting set problem. In the SHSP, we are given a supergraph \(H = (V, \mathcal{C})\) and a nonnegative submodular function on the set \(2^{V}\). The objective is to determine a vertex subset to cover all hyperedges such that the cost of submodular covering is minimized. Our main work is to present a rounding algorithm and a primal-dual algorithm respectively for the SHSP and prove that they both have the approximation ratio \(k\), where \(k\) is the maximum number of vertices in all hyperedges.

In recent years, there is a lot of interest in the topic of conveying the groups of planar graphs with an unvarying metric dimension. A few types of planar graphs have recently had their locating number (or metric dimension) determined, and an outstanding problem concerning these graphs was brought up that: Illustrate the types of planar graphs \(\Upsilon\) that can be generated from a graph \(\Phi\) through the addition of more edges to \(\Phi\), such that \(dim(\Phi)=dim(\Upsilon)\) and \(\mathbb{V}(\Phi)=\mathbb{V}(\Upsilon)\). While proceeding in a similar directives, we identify two families of radially identical planar graphs with unaltered metric dimension in this study: \(\digamma_{n,m}\) and \(\gimel_{n,m}\). We do this by establishing that \(dim(\digamma_{n,m})=dim(\gimel_{n,m})\) and \(\mathbb{V}(\digamma_{n,m})=\mathbb{V}(\gimel_{n,m})\), respectively. We acquire another family of a radially symmetrical plane graph (i.e., \(\daleth_{n,m}\)) with a constant metric dimension. We show that all the vertices of these classes of the plane graphs can potentially be identified with just three well-chosen nodes.

The purpose of this work was to use machine learning classification models and hyperspectral camera technologies to create a model of surface damage to garlic. 140 of the 184 garlic plants of which 44 were used for test validation were pre-treated for surface damage. First, we examined the data in ENVI under various damage scenarios using the normalised vegetation index (NDVI) approach. 579 pixels were then chosen for the training of the logistic regression model. Finally, we used 54 garlic bulbs to practically validate the model. Although tiny regions could not be precisely identified, the mouldy portion of the garlic’s surface could be identified using the NDVI technique. 90% accuracy was attained using the 90% classification model constructed using the logistic regression approach. Garlic’s surface damage, even at first mild ones, was correctly identified. The creation of this model for identifying garlic damage lowers the cost of detecting garlic damage and broadens the use of hyperspectral technologies in garlic detection.

In this paper, the sensor is applied to the collection of rock parameter data. Aiming at the classification and evaluation of stability (i.e. rock quality), an attribute recognition model for the classification and evaluation of surrounding rock quality of underground engineering is established. Using multi-source data fusion and orthogonal numerical simulation test methods, the effects of rock mechanics parameters on the horizontal convergence of the tunnel, the settlement of the vault and the plastic zone coefficient are studied. Six factors (elastic modulus, Poisson’s ratio, internal friction angle, tensile strength, cohesion and density) and three levels of orthogonal experimental solutions were selected. The method of defining similar weight by using similar number to determine the weight of evaluation index, so as to calculate the comprehensive attribute measure, and apply confidence criteria to identify the stability of rock samples. Through the analysis and evaluation of rock mass quality classification of underground engineering, the application of the model and the evaluation method of rock mass quality classification are explained. The test results match the orthogonal test results; Considering the stability of tunnel envelope, the horizontal convergence, vault settlement and plastic zone coefficient after excavation should be comprehensively considered.

This paper proposes a comprehensive framework for testing and evaluating automatic ambulances, crucial for ensuring their reliability and safety in real-world scenarios. The framework includes designing test scenarios with varying complexity, covering environmental factors like road conditions, weather, and obstacles. An evaluation index system is introduced, comprising driving security, ride comfort, intelligence, and efficiency. Methodologies for calculating indicator weights, using the CRITIC and AHP methods, are presented to ensure fair evaluation. Additionally, evaluation methods including qualitative and quantitative techniques, such as grey correlation theory, are discussed. The test results show that the assessment results of the traditional fuzzy comprehensive evaluation method and the grey correlation theory evaluation method are highly consistent. The change in vehicle speed has less of an effect on accuracy during the real-time assessment process when the time interval is set to 0.1s, and the evaluation time of 0.098s can satisfy the requirement that the planning time of autonomous driving vehicles be shorter than 200 ms.

In recent years, the use of smart data analysis method to predict the stock price is financial technology; important issues in the field of finch. However, there are many technical indicators and human subjective factors will affect the stock price forecast, so we must effectively grasp the important influence indicators to improve the accuracy of stock price forecast. Therefore, this study uses four machine learning algorithms to predict and analyze the stock price fluctuation through the screening process of technical indicators, and then selects the important technical indicators. In addition, due to the uncertainty and fuzziness of the attributes of technical indicators and human subjective judgment, this study uses the fuzzy inference method to construct the fuzzy inference system to predict the rise and fall of stock price, and proposes the prediction method of the range of the rise and fall of stock price. Finally, this paper makes an empirical analysis on the stock price data of three companies. The results show that the accuracy of stock price forecast is more than 82.13%, and the average accuracy of stock price forecast is more than 83%. Therefore, the fuzzy inference prediction system proposed in this study not only has the theoretical basis, but also can effectively predict the trend and range of stock price, which has practical value and contribution to investors.

Removing clouds is an essential preprocessing step in analyzing remote sensing images, as cloud-based overlays commonly occur in optical remote sensing images and can significantly limit the usability of the acquired data. Deep learning has exhibited remarkable progress in remote sensing, encompassing scene classification and change detection tasks. Nevertheless, the appli-cation of deep learning techniques to cloud removal in remote sensing images is currently con-strained by the limited availability of training datasets explicitly tailored for neural networks. This paper presents the Remote sensing Image Cloud rEmoving dataset (RICE) to address this challenge and proposes baseline models incorporating a convolutional attention mechanism, which has demonstrated superior performance in identifying and restoring cloud-affected regions, with quantitative results indicating a 3.08% improvement in accuracy over traditional methods. This mechanism empowers the network to comprehend better the spatial structure, local details, and inter-channel correlations within remote sensing images, thus effectively addressing the diverse distributions of clouds. Moreover, by integrating this attention mechanism, our models achieve a crucial comparison advantage, outperforming existing state-of-the-art techniques in terms of both visual quality and quantitative metrics. We propose adopting the Learned Per-ceptual Image Patch Similarity metric, which emphasizes perceptual similarity, to evaluate the quality of cloud-free images generated by the models. Our work not only contributes to advancing cloud removal techniques in remote sensing but also provides a comprehensive evaluation framework for assessing the fidelity of the generated images.

Let \(g,f:V(G)\rightarrow\{0,1,2,3,\cdots\}\) be two functions satisfying \(g(x)\leq f(x)\) for every \(x\in V(G)\). A \((g,f)\)-factor of \(G\) is
defined as a spanning subgraph \(F\) of \(G\) such that \(g(x)\leq d_F(x)\leq f(x)\) for every \(x\in V(G)\). An \((f,f)\)-factor is simply called
an \(f\)-factor. Let \(\varphi\) be a nonnegative integer-valued function defined on \(V(G)\). Set
\[
D_{g,f}^{even}=\Big\{\varphi: g(x)\leq\varphi(x)\leq f(x) \text{ for every } x\in V(G) \text{ and } \sum\limits_{x\in V(G)}\varphi(x) \text{ is even}\Big\}.
\]
If for each \(\varphi\in D_{g,f}^{even}\), \(G\) admits a \(\varphi\)-factor, then we say that \(G\) admits all \((g,f)\)-factors. All \((g,f)\)-factors
are said to be all \([1,k]\)-factors if \(g(x)\equiv1\) and \(f(x)\equiv k\) for any \(x\in V(G)\). In this paper, we verify that for a connected multigraph
\(G\) satisfying \(N_G(X)=V(G) \text{ or } |N_G(X)|>\Big(1+\frac{1}{k+1}\Big)|X|-1\) for every \(X\subset V(G)\), \(kG\) admits all \([1,k]\)-factors, where
\(k\geq2\) is an integer and \(kG\) denotes the graph derived from \(G\) by replacing every edge of \(G\) with \(k\) parallel edges.

We consider finitely presented groups \(G_{mn}\) as follows:
\[
G_{mn}=\left\langle x, y \mid {x^{m}}={y^{n}}=1, {[x, y]^{x}}=[x, y], {[x, y]^{y}}=[x, y] \right\rangle m, n\ge 2.
\]
In this paper, we first study the groups \(G_{mn}\). Then by using the properties of \(G_{mm}\) and \(t-\)Fibonacci sequences in
finitely generated groups, we show that the period of \(t-\)Fibonacci sequences in \(G_{mm}\) are a multiple of \(K(t, m)\). In particular for \(t \geq 3\) and \(p=2\), we prove \({{LEN}_{t}({{G}_{pp}})}= 2K(t,p)\).

In this paper, we consider a degree sum condition sufficient to imply the existence of \(k\) vertex-disjoint chorded cycles in a graph \(G\).
Let \(\sigma_4(G)\) be the minimum degree sum of four independent vertices of \(G\).
We prove that if \(G\) is a graph of order at least \(11k+7\) and \(\sigma_4(G)\ge 12k-3\) with \(k\ge 1\),
then \(G\) contains \(k\) vertex-disjoint chorded cycles.
We also show that the degree sum condition on \(\sigma_4(G)\) is sharp.

This study addresses challenges in rural planning amid economic growth and the implementation of rural revitalization policies. The aim is to enhance the integration of cultural and ecological elements in rural areas, combating issues such as the fading village atmosphere and incomplete agricultural chains. The research focuses on optimizing the random forest algorithm to explore innovative approaches to landscape planning and design for rural human settlements. Using the moving window method, the study computes two-dimensional and three-dimensional landscape indices in surrounding villages of Beijing, conducting a multi-scale analysis of the living environment. Power function fitting indicates an optimal window size of approximately 700 meters for studying the relationship between art patterns and three-dimensional landscape patterns in the rural area. The findings offer insights into improving rural living environments through effective landscape planning and design influenced by artistic modes.

In the major cities with many high-rise buildings in contemporary China, land resources are becoming more and more scarce, and the urban ecological environment is in urgent need of recycling, and due to the blind imitation of Western culture and design mode and the neglect of China traditional regional culture, the urban landscape lacks interaction, resonance, and sense of belonging with citizens, and the phenomenon of landscape similarity emerges in various cities, focusing on the landscape space of urban complexes. There are also these problems. Urban residents urgently need a third space that can adjust their physical, mental, and spiritual needs. How to design an urban complex landscape that meets the aesthetic needs and humanistic needs of contemporary cities and has regional characteristics has become the first important task of my research. Folk art is an artistic treasure created by the working people in their production and life. Folk art is the embodiment of cultural regionality and the foundation of national culture. It awakens people’s awareness of the importance of local culture, awakens people’s sense of belonging, and is closer to the local public life. Today, the living soil and social and humanistic environment of folk art are in the process of urbanization in China, and there is a trend of gradual disappearance of lifestyle changes. How to make the contemporary urban complex landscape an organic soil for the survival, expression, application, and development of folk art is an important task in contemporary urban landscape design. Based on optimization, related concepts such as symbols, folk art symbols, urban complexes, urban complex landscape design, etc. have been sorted out. The relevant experimental results show that the construction land accuracy of the logistic regression model based on genetic algorithm has increased from 78.0% to 85.3%. kappa increased from 74.5% to 81.2%. Research shows that the logistic regression parameter optimization method based on genetic algorithm has better simulation effect than the conventional logistic regression method and is more suitable for the situation of unbalanced data distribution and many data features in the simulation of large-scale urban land dynamic changes.

Digital services, including healthcare, among others, have recently seen a massive volume of complicated data that arrives rapidly due to a rapid increase in the number of smart devices, focusing on the needs of regional emerging economic development and industrial structure adjustment, this paper explores the dynamic adjustment of a major in which schools, governments, enterprises and international cooperation participate in the development of regional emerging economies. mechanism. Based on the concept of future-oriented development, formulate a development plan for the legal profession, build a community of government, school, enterprise and international cooperation, promote the vigorous implementation of engineering practice education, and cultivate high-quality, high-level, international graduates, and to form the school-running characteristics of law majors in local application-oriented undergraduate colleges.

Given a connected simple undirected graph \(G=(V,E)\), a subset \(S\) of \(V\) is \(P_3\)-convex if each vertex of \(G\) not in \(S\) has at most one neighbor in \(S\). The \(P_3\)-convex hull \(\langle S\rangle\) of \(S\) is the smallest \(P_3\)-convex set containing \(S\). A Carathéodory set of \(G\) is a set \(S\subseteq V\) such that \(\langle S \rangle \setminus \bigcup_{w\in S} \langle S\setminus \{w\} \rangle\) is non-empty. The Carathéodory number of \(G\), denoted by \(C(G)\), is the largest cardinality of a Carathéodory set of \(G\). In this paper, we settle the conjecture posed by Barbosa et al. appeared in [SIAM J. Discrete Math. 26 (2012) 929–939] in the affirmative, which states that for a claw-free graph \(G\) of order \(n(G)\), the Carathéodory number \(C(G)\) of the \(P_3\)-convexity satisfies \(C(G) \leq \frac{2n(G)+6}{5}\). Furthermore, we determine all graphs attaining the bound.

The key factor that promotes Vocational students development is the development of movement, which requires children to have excellent motor skills to develop their intellectual level, physical function and social adaptability in daily study and life. Data mining technology is an economical and practical core technology, which obtains useful information for the service system from a large amount of data. However, many teaching deficiencies in this area are prevalent in the field of early childhood education. In the current research on the content of Vocational students physical activities, a large amount of data information needs the support of data mining technology. This paper aims at how to combine data mining technology to study the content of Vocational students sports activities from the perspective of movement development, establish a decision support system for Vocational students sports activities, and conduct experiments on Vocational students sports activities from the perspective of action scientific arrangement and implementation of activity content, carry out empirical research on the content of Vocational students sports activities.

When using airborne LiDAR point clouds for city modelling and road extraction, point cloud classification is a crucial step. There are numerous ways for classifying point clouds, but there are still issues like redundant multi-dimensional feature vector data and poor point cloud classification in intricate situations. A point cloud classification method built on the fusing of multikernel feature vectors is suggested as a solution to these issues. The technique employs random forest to classify point cloud data by merging colour information, and it extracts feature vectors based on point primitives and object primitives, respectively. In this study, a densely populated area was chosen as the study area. Light airborne LIDAR mounted on a delta wing was used to collect point cloud data at a low altitude (170 m) over a dense cross-course. The point cloud data were then combined, corrected, and enhanced with texture data, and the houses were vectorized on the point cloud. The accuracy of the results was then assessed. With a median inaccuracy of 4.8 cm and a point cloud data collection rate of 83.3%, using airborne LIDAR to measure house corners can significantly lighten the labour associated with external house corner measurements.This test extracts the texture information of point cloud data through the efficient processing of high-density point cloud data, providing a reference for the application of texture information of airborne LIDAR data and a clear understanding of its accuracy.

One of the fundamental properties of the hypercube \( Q_n \) is that it is bipancyclic as \( Q_n \) has a cycle of length \( l \) for every even integer \( l \) with \( 4 \leq l \leq 2^n \). We consider the following problem of generalizing this property: For a given integer \( k \) with \( 3 \leq k \leq n \), determine all integers \( l \) for which there exists an \( l \)-vertex, \( k \)-regular subgraph of \( Q_n \) that is both \( k \)-connected and bipancyclic. The solution to this problem is known for \( k = 3 \) and \( k = 4 \). In this paper, we solve the problem for \( k = 5 \). We prove that \( Q_n \) contains a \( 5 \)-regular subgraph on \( l \) vertices that is both \( 5 \)-connected and bipancyclic if and only if \( l \in \{32, 48\} \) or \( l \) is an even integer satisfying \( 52 \leq l \leq 2^n \). For general \( k \), we establish that every \( k \)-regular subgraph of \( Q_n \) has \( 2^k, 2^k + 2^{k-1} \) or at least \( 2^k + 2^{k-1} + 2^{k-3} \) vertices.

Coded caching technology can better alleviate network traffic congestion. Since many of the centralized coded caching schemes now in use have high subpacketization, which makes scheme implementation more challenging, coded caching schemes with low subpacketization offer a wider range of practical applications. It has been demonstrated that the coded caching scheme can be achieved by creating a combinatorial structure named placement delivery array (PDA). In this work, we employ vector space over a finite field to obtain a class of PDA, calculate its parameters, and consequently achieve a coded caching scheme with low subpacketization. Subsequently, we acquire a new MN scheme and compare it with the new scheme developed in this study. The subpacketization \(F\) of the new scheme has significant advantages. Lastly, the number of users \(K\), cache fraction \(\frac{M}{N}\), and subpacketization \(F\) have advantages to some extent at the expense of partial transmission rate \(R\) when compared to the coded caching scheme in other articles.

We continue the study of Token Sliding (reconfiguration) graphs of independent sets initiated by the authors in an earlier paper [Graphs Comb. 39.3, 59, 2023]. Two of the topics in that paper were to study which graphs \(G\) are Token Sliding graphs and which properties of a graph are inherited by a Token Sliding graph. In this paper, we continue this study specializing in the case of when \(G\) and/or its Token Sliding graph \(\mathsf{TS}_k(G)\) is a tree or forest, where \(k\) is the size of the independent sets considered. We consider two problems. The first is to find necessary and sufficient conditions on \(G\) for \(\mathsf{TS}_k(G)\) to be a forest. The second is to find necessary and sufficient conditions for a tree or forest to be a Token Sliding graph. For the first problem, we give a forbidden subgraph characterization for the cases of \(k=2,3\). For the second problem, we show that for every \(k\)-ary tree \(T\) there is a graph \(G\) for which \(\mathsf{TS}_{k+1}(G)\) is isomorphic to \(T\). A number of other results are given along with a join operation that aids in the construction of \(\mathsf{TS}_k\)-graphs.

In this paper, we introduce graceful and near graceful labellings of several families of windmills. In particular, we use Skolem-type sequences to prove (near) graceful labellings exist for windmills with \(C_{3}\) and \(C_{4}\) vanes, and infinite families of \(3,5\)-windmills and \(3,6\)-windmills. Furthermore, we offer a new solution showing that the graph obtained from the union of \(t\) 5-cycles with one vertex in common (\(C_{5}^{t}\)) is graceful if and only if \(t \equiv 0, 3 \pmod{4}\) and near graceful when \(t \equiv 1, 2 \pmod{4}\).

We study groups generated by sets of pattern avoiding permutations. In the first part of the paper, we prove some general results concerning the structure of such groups. In particular, we consider the sequence \((G_n)_{n \geq 0}\), where \(G_n\) is the group generated by a subset of the symmetric group \(S_n\) consisting of permutations that avoid a given set of patterns. We analyze under which conditions the sequence \((G_n)_{n \geq 0}\) is eventually constant. Moreover, we find a set of patterns such that \((G_n)_{n \geq 0}\) is eventually equal to an assigned symmetric group. Furthermore, we show that any non-trivial simple group cannot be obtained in this way and describe all the non-trivial abelian groups that arise in this way. In the second part of the paper, we carry out a case-by-case analysis of groups generated by permutations avoiding a few short patterns.

We consider the eccentric graph of a graph \(G\), denoted by \(\mathrm{ecc}(G)\), which has the same vertex set as \(G\), and two vertices in the eccentric graph are adjacent if and only if their distance in \(G\) is equal to the eccentricity of one of them. In this paper, we present a fundamental requirement for the isomorphism between \(\mathrm{ecc}(G)\) and the complement of \(G\), and show that the previous necessary condition given in the literature is inadequate. Also, we obtain that the diameter of \(\mathrm{ecc}(T)\) is at most 3 for any tree and get some characterizations of the eccentric graph of trees.

Let \(G\) be a finite simple undirected \((p, q)\)-graph, with vertex set \(V(G)\) and edge set \(E(G)\) such that \(p = |V(G)|\) and \(q = |E(G)|\). A super edge-magic total labeling \(f\) of \(G\) is a bijection \(f \colon V(G) \cup E(G) \longrightarrow \{1, 2, \dots, p+q\}\) such that for all edges \(uv \in E(G)\), \(f(u) + f(v) + f(uv) = c(f)\), where \(c(f)\) is called a magic constant, and \(f(V(G)) = \{1, \dots, p\}\). The minimum of all \(c(f)\), where the minimum is taken over all the super edge-magic total labelings \(f\) of \(G\), is defined to be the super edge-magic total strength of the graph \(G\). In this article, we work on certain classes of unicyclic graphs and provide evidence to conjecture that the super edge-magic total strength of a certain family of unicyclic \((p, q)\)-graphs is equal to \(2q + \frac{n+3}{2}\).

For a poset \(P = C_a \times C_b\), a subset \(A \subseteq P\) is called a chain blocker for \(P\) if \(A\) is inclusion-wise minimal with the property that every maximal chain in \(P\) contains at least one element of \(A\), where \(C_i\) is the chain \(1 < \cdots < i\). In this article, we define the shelter of the poset \(P\) to give a complete description of all chain blockers of \(C_5 \times C_b\) for \(b \geq 1\).

This project aims at investigating properties of channel detecting codes on specific domains \(1^+0^+\). We focus on the transmission channel with deletion errors. Firstly, we discuss properties of channels with deletion errors. We propose a certain kind of code that is a channel detecting (abbr. \(\gamma\)-detecting) code for the channel \(\gamma = \delta(m, N)\) where \(m < N\). The characteristic of this \(\gamma\)-detecting code is considered. One method is provided to construct \(\gamma\)-detecting code. Finally, we also study a kind of special channel code named \(\tau(m, N)\)-srp code.

A chemical structure specifies the molecular geometry of a given molecule or solid in the form of atom arrangements. One way to analyze its properties is to simulate its formation as a product of two or more simpler graphs. In this article, we take this idea to find upper and lower bounds for the generalized Randić index \(\mathcal{R}_{\alpha}\) of four types of graph products, using combinatorial inequalities. We finish this paper by providing the bounds for \(\mathcal{R}_{\alpha}\) of a line graph and rooted product of graphs.

Let \(G\) be a \((p, q)\) graph. Let \(f: V(G) \to \{1, 2, \ldots, k\}\) be a map where \(k \in \mathbb{N}\) is a variable and \(k > 1\). For each edge \(uv\), assign the label \(\gcd(f(u), f(v))\). \(f\) is called \(k\)-Total prime cordial labeling of \(G\) if \(\left|t_{f}(i) – t_{f}(j)\right| \leq 1\), \(i, j \in \{1, 2, \ldots, k\}\) where \(t_{f}(x)\) denotes the total number of vertices and edges labeled with \(x\). A graph with a \(k\)-total prime cordial labeling is called \(k\)-total prime cordial graph. In this paper, we investigate the 4-total prime cordial labeling of some graphs like dragon, Möbius ladder, and corona of some graphs.

Let \(G = (V, E)\) be a graph with vertex set \(V\) and edge set \(E\). An edge labeling \(f: E \to Z_{2}\) induces a vertex labeling \(f^{+} : V \to Z_{2}\) defined by \( f^{+}(v) \equiv \sum_{uv \in E} f(uv) \pmod 2 \), for each vertex \(v \in V\). For \(i \in Z_{2}\), let \( v_{f}(i) = |\{v \in V : f^+(v) = i\}| \) and \( e_{f}(i) = |\{e \in E : f(e) = i\}| \). An edge labeling \(f\) of a graph \(G\) is said to be edge-friendly if \( |e_{f}(1) – e_{f}(0)| \le 1 \). The set \(\{v_f(1) – v_f(0) : f \text{ is an edge-friendly labeling of } G\}\) is called the full edge-friendly index set of \(G\). In this paper, we shall determine the full edge-friendly index sets of one point union of cycles.

After the Chartrand definition of graph labeling, since 1988 many graph families have been labeled through mathematical techniques. A basic approach in those labelings was to find a pattern among the labels and then prove them using sequences and series formulae. In 2018, Asim applied computer-based algorithms to overcome this limitation and label such families where mathematical solutions were either not available or the solution was not optimum. Asim et al. in 2018 introduced the algorithmic solution in the area of edge irregular labeling for computing a better upper-bound of the complete graph \(es(K_n)\) and a tight upper-bound for the complete \(m\)-ary tree \({es(T}_{m,h})\) using computer-based experiments. Later on, more problems like complete bipartite and circulant graphs were solved using the same technique. Algorithmic solutions opened a new horizon for researchers to customize these algorithms for other types of labeling and for more complex graphs. In this article, to compute edge irregular \(k\)-labeling of star \(S_{m,n}\) and banana tree \({BT}_{m,n}\), new algorithms are designed, and results are obtained by executing them on computers. To validate the results of computer-based experiments with mathematical theorems, inductive reasoning is adopted. Tabulated results are analyzed using the law of double inequality and it is concluded that both families of trees observe the property of edge irregularity strength and are tight for \(\left\lceil \frac{|V|}{2} \right\rceil\)-labeling.

A graph \(G\) is called a fractional ID-\((g,f)\)-factor-critical covered graph if for any independent set \(I\) of \(G\) and for every edge \(e \in E(G-I)\), \(G-I\) has a fractional \((g,f)\)-factor \(h\) such that \(h(e) = 1\). We give a sufficient condition using degree condition for a graph to be a fractional ID-\((g,f)\)-factor-critical covered graph. Our main result is an extension of Zhou, Bian, and Wu’s previous result [S. Zhou, Q. Bian, J. Wu, A result on fractional ID-\(k\)-factor-critical graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 87(2013) 229–236] and Yashima’s previous result [T. Yashima, A degree condition for graphs to be fractional ID-\([a,b]\)-factor-critical, Australasian Journal of Combinatorics 65(2016) 191–199].

In this article, we define \(q\)-generalized Fibonacci polynomials and \(q\)-generalized Lucas polynomials using \(q\)-binomial coefficient and obtain their recursive properties. In addition, we introduce generalized \(q\)-Fibonacci matrix and generalized \(q\)-Lucas matrix, then we derive their basic identities. We define \((k,q,t)\)-symmetric generalized Fibonacci matrix and \((k,q,t)\)-symmetric generalized Lucas matrix, then we give the Cholesky factorization of these matrices. Finally, we give determinantal and permanental representations of these new polynomial sequences.