For a graph \(G\), two vertices \(x,y\in G\) are said to be resolved by a vertex \(s\in G\), if \(d(x|s)\neq d(y|s)\). The minimum cardinality of such a resolving set \(\textit{R}\) in \(G\) is called its metric dimension. A resolving set \(\textit{R}\) is said to be fault-tolerant, if for every \(p\in R\), \(R-p\) preserves the property of being a resolving set. A fault-tolerant metric dimension of \(G\) is a minimal possible order fault-tolerant resolving set. A wide variety of situations, in which connection, distance, and connectivity are important aspects, call for the utilisation of metric dimension. The structure and dynamics of complex networks, as well as difficulties connected to robotics network design, navigation, optimisation, and facility positioning, are easier to comprehend as a result of this. As a result of the relevant concept of metric dimension, robots are able to optimise their methods of localization and navigation by making use of a limited number of reference locations. As a consequence of this, numerous applications of robotics, including collaborative robotics, autonomous navigation, and environment mapping, have become more precise, efficient, and resilient. The arithmetic graph \(A_l\) is defined as the graph with its vertex set as the set of all divisors of \(l\), where \(l\) is a composite number and \(l = p^{\gamma_1}_{1} p^{\eta_2}_{2}, \dots, p^{n}_{n}\), where \(p_n \geq 2\) and the \(p_i\)’s are distinct primes. Two distinct divisors \(x, y\) of \(l\) are said to have the same parity if they have the same prime factors (i.e., \(x = p_{1}p_{2}\) and \(y = p^{2}_{1}p^{3}_{2}\) have the same parity). Further, two distinct vertices \(x, y \in A_l\) are adjacent if and only if they have different parity and \(\gcd(x, y) = p_i\) (greatest common divisor) for some \(i \in \{1, 2, \dots, t\}\). This article is dedicated to the investigation of the arithmetic graph of a composite number \(l\), which will be referred to throughout the text as \(A_{l}\). In this study, we compute the fault-tolerant resolving set and the fault-tolerant metric dimension of the arithmetic graph \(A_{l}\), where \(l\) is a composite number.

In this paper, we identify LWO graphs, f\-ind the minimum \(\lambda\) for decomposition of \(\lambda K_n\) into these graphs, and show that for all viable values of \(\lambda\), the necessary conditions are suf\-f\-icient for LWO–decompositions using cyclic decompositions from base graphs.

For a graph \( G \) and a subgraph \( H \) of a graph \( G \), an \( H \)-decomposition of the graph \( G \) is a partition of the edge set of \( G \) into subsets \( E_i \), \( 1 \leq i \leq k \), such that each \( E_i \) induces a graph isomorphic to \( H \). In this paper, it is proved that every simple connected unicyclic graph of order five decomposes the \( \lambda \)-fold complete equipartite graph whenever the necessary conditions are satisfied. This generalizes a result of Huang, *Utilitas Math.* 97 (2015), 109–117.

We classify the geometric hyperplanes of the Segre geometries, that is, direct products of two projective spaces. In order to do so, we use the concept of a generalised duality. We apply the classification to Segre varieties and determine precisely which geometric hyperplanes are induced by hyperplanes of the ambient projective space. As a consequence we find that all geometric hyperplanes are induced by hyperplanes of the ambient projective space if, and only if, the underlying field has order \(2\) or \(3\).

A modification of Merino-Mǐcka-Mütze’s solution to a combinatorial generation problem of Knuth is proposed in this survey. The resulting alternate form to such solution is compatible with a reinterpretation by the author of a proof of existence of Hamilton cycles in the middle-levels graphs. Such reinterpretation is given in terms of a dihedral quotient graph associated to each middle-levels graph. The vertices of such quotient graph represent Dyck words and their associated ordered trees. Those Dyck words are linearly ordered via a rooted tree that covers all their tight, or irreducible, forms, offering an universal reference point of view to express and integrate the periodic paths, or blocks, whose concatenation leads to Hamilton cycles resulting from the said solution.

The hub cover pebbling number, \(h^{*}(G)\), of a graph $G$, is the least non-negative integer such that from all distributions of \(h^{*}(G)\) pebbles over the vertices of \(G\), it is possible to place at least one pebble each on every vertex of a set of vertices of a hub set for \(G\) using a sequence of pebbling move operations, each pebbling move operation removes two pebbles from a vertex and places one pebble on an adjacent vertex. Here we compute the hub cover pebbling number for wheel related graphs.

An outer independent double Roman dominating function (OIDRDF) on a graph \( G \) is a function \( f : V(G) \to \{0, 1, 2, 3\} \) having the property that (i) if \( f(v) = 0 \), then the vertex \( v \) must have at least two neighbors assigned 2 under \( f \) or one neighbor \( w \) with \( f(w) = 3 \), and if \( f(v) = 1 \), then the vertex \( v \) must have at least one neighbor \( w \) with \( f(w) \ge 2 \) and (ii) the subgraph induced by the vertices assigned 0 under \( f \) is edgeless. The weight of an OIDRDF is the sum of its function values over all vertices, and the outer independent double Roman domination number \( \gamma_{oidR}(G) \) is the minimum weight of an OIDRDF on \( G \). The \( \gamma_{oidR} \)-stability (\( \gamma^-_{oidR} \)-stability, \( \gamma^+_{oidR} \)-stability) of \( G \), denoted by \( {\rm st}_{\gamma_{oidR}}(G) \) (\( {\rm st}^-_{\gamma_{oidR}}(G) \), \( {\rm st}^+_{\gamma_{oidR}}(G) \)), is defined as the minimum size of a set of vertices whose removal changes (decreases, increases) the outer independent double Roman domination number. In this paper, we determine the exact values on the \( \gamma_{oidR} \)-stability of some special classes of graphs, and present some bounds on \( {\rm st}_{\gamma_{oidR}}(G) \). In addition, for a tree \( T \) with maximum degree \( \Delta \), we show that \( {\rm st}_{\gamma_{oidR}}(T) = 1 \) and \( {\rm st}^-_{\gamma_{oidR}}(T) \le \Delta \), and characterize the trees that achieve the upper bound.

We introduce a two-player game where the goal is to illuminate all edges of a graph. At each step the first player, called Illuminator, taps a vertex. The second player, called Adversary, reveals the edges incident with that vertex (consistent with the edges incident with the already tapped vertices). Illuminator tries to minimize the taps needed, and the value of the game is the number of taps needed with optimal play. We provide bounds on the value in trees and general graphs. In particular, we show that the value for the path on \( n \) vertices is \( \frac{2}{3} n + O(1) \), and there is a constant \( \varepsilon > 0 \) such that for every caterpillar on \( n \) vertices, the value is at most \( (1 – \varepsilon) n + 1 \).

Let \(G\) be a group, and let \(c\in\mathbb{Z}^+\cup\{\infty\}\). We let \(\sigma_c(G)\) be the maximal size of a subset \(X\) of \(G\) such that, for any distinct \(x_1,x_2\in X\), the group \(\langle x_1,x_2\rangle\) is not \(c\)-nilpotent; similarly we let \(\Sigma_c(G)\) be the smallest number of \(c\)-nilpotent subgroups of \(G\) whose union is equal to \(G\). In this note we study \(D_{2k}\), the dihedral group of order \(2k\). We calculate \(\sigma_c(D_{2k})\) and \(\Sigma_c(D_{2k})\), and we show that these two numbers coincide for any given \(c\) and \(k\).

Let \(p > 2\) be prime and \(r \in \{1,2, \ldots, p-1\}\). Denote by \(c_{p}(n)\) the number of \(p\)-regular partitions of \(n\) in which parts can occur not more than three times. We prove the following: If \(8r + 1\) is a quadratic non-residue modulo \(p\), \(c_{p}(pn + r) \equiv 0 \pmod{2}\) for all nonnegative integers \(n\).

Let \( G=(V,E) \) be a simple connected graph with vertex set \( G \) and edge set \( E \). The harmonic index of graph \( G \) is the value \( H(G)=\sum_{uv\in E(G)} \frac{2}{d_u+d_v} \), where \( d_x \) refers to the degree of \( x \). We obtain an upper bound for the harmonic index of trees in terms of order and the total domination number, and we characterize the extremal trees for this bound.

A \((p, g)\)-graph \(G\) is Euclidean if there exists a bijection \(f: V \to \{1, 2, \ldots, p\}\) such that for any induced \(C_3\)-subgraph \(\{v_1, v_2, v_3\}\) in \(G\) with \(f(v_1) < f(v_2) < f(v_3)\), we have that \(f(v_1) + f(v_2) > f(v_3)\). The Euclidean Deficiency of a graph \(G\) is the smallest integer \(k\) such that \(G \cup N_k\) is Euclidean. We study the Euclidean Deficiency of one-point union and one-edge union of complete graphs.

The dominating set of a graph \(G\) is a set of vertices \(D\) such that for every \(v \in V(G)\) either \(v \in D\) or \(v\) is adjacent to a vertex in \(D\). The domination number, denoted \(\gamma(G)\), is the minimum number of vertices in a dominating set. In 1998, Haynes and Slater [1] introduced paired-domination. Building on paired-domination, we introduce 3-path domination. We define a 3-path dominating set of \(G\) to be \(D = \{ Q_1,Q_2,\dots , Q_k\, |\:Q_i \text{ is a 3-path}\}\) such that the vertex set \(V(D) = V(Q_1) \cup V(Q_2) \cup \dots \cup V(Q_k)\) is a dominating set. We define the 3-path domination number, denoted by \(\gamma_{P_3}(G)\), to be the minimum number of 3-paths needed to dominate \(G\). We show that the 3-path domination problem is NP-complete. We also prove bounds on \(\gamma_{P_3}(G)\) and improve those bounds for particular families of graphs such as Harary graphs, Hamiltonian graphs, and subclasses of trees. In general, we prove \(\gamma_{P_3}(G) \leq \frac{n}{3}\).

Two colorings have been introduced recently where an unrestricted coloring \(c\) assigns nonempty subsets of \([k]=\{1,\ldots,k\}\) to the edges of a (connected) graph \(G\) and gives rise to a vertex-distinguishing vertex coloring by means of set operations. If each vertex color is obtained from the union of the incident edge colors, then \(c\) is referred to as a strong royal coloring. If each vertex color is obtained from the intersection of the incident edge colors, then \(c\) is referred to as a strong regal coloring. The minimum values of \(k\) for which a graph \(G\) has such colorings are referred to as the strong royal index of \(G\) and the strong regal index of \(G\) respectively. If the induced vertex coloring is neighbor distinguishing, then we refer to such edge colorings as royal and regal colorings. The royal chromatic number of a graph involves minimizing the number of vertex colors in an induced vertex coloring obtained from a royal coloring. In this paper, we provide new results related to these two coloring concepts and establish a connection between the corresponding chromatic parameters. In addition, we establish the royal chromatic number for paths and cycles.

A ranking on a graph \(G\) is a function \(f: V(G) \rightarrow \left\{1, 2, \ldots, k \right\}\) with the following restriction: if \(f(u)=f(v)\) for any \(u, v \in V(G)\), then on every \(uv\) path in \(G\), there exists a vertex \(w\) with \(f(w) > f(u)\). The optimality of a ranking is conventionally measured in terms of the \(l_{\infty}\) norm of the sequence of labels produced by the ranking. In \cite{jacob2017lp} we compared this conventional notion of optimality with the \(l_p\) norm of the sequence of labels in the ranking for any \(p \in [0,\infty)\), showing that for any non-negative integer \(c\) and any non-negative real number \(p\), we can find a graph such that the sets of \(l_p\)-optimal and \(l_{\infty}\)-optimal rankings are disjoint. In this paper we identify some graphs whose set of \(l_p\)-optimal rankings and set of \(l_{\infty}\)-optimal rankings overlap. In particular, we establish that for paths and cycles, if \(p>0\) then \(l_p\) optimality implies \(l_{\infty}\) optimality but not the other way around, while for any complete multipartite graph, \(l_p\) optimality and \(l_{\infty}\) optimality are equivalent.

We use a representation for the spanning tree where a parent function maps non-root vertices to vertices. Two spanning trees are defined to be adjacent if their function representations differ at exactly one vertex. Given a graph \(G\), we show that the graph \(H\) with all spanning trees of \(G\) as vertices and any two vertices being adjacent if and only if their parent functions differ at exactly one vertex is connected.

A \((0,1)\)-labeling of a set is said to be friendly if the number of elements of the set labeled 0 and the number labeled 1 differ by at most 1. Let \(g\) be a labeling of the edge set of a graph that is induced by a labeling \(f\) of the vertex set. If both \(g\) and \(f\) are friendly then \(g\) is said to be a cordial labeling of the graph. We extend this concept to directed graphs and investigate the cordiality of directed graphs. We show that all directed paths and all directed cycles are cordial. We also discuss the cordiality of oriented trees and other digraphs.

We propose and study the problem of finding the smallest nonnegative integer \(s\) such that a GDD\((m, n, 3; 0, \lambda)\) can be embedded into a BIBD\((mn + s, 3, \lambda)\). We find the values of \(s\) for all cases except for the case where \(n \equiv 5 \pmod{6}\) and \(m \equiv 1, 3 \pmod{6}\) and \(m \ge 3\), which remains as an open problem.

A simple graph \(G\) with \(p\) vertices is said to be vertex-Euclidean if there exists a bijection \(f: V(G) \rightarrow \{1, 2, \ldots, p\}\) such that \(f(v_1) + f(v_2) > f(v_3)\) for each \(C_3\)-subgraph with vertex set \(\{v_1, v_2, v_3\}\), where \(f(v_1) < f(v_2) < f(v_3)\). More generally, the vertex-Euclidean deficiency of a graph \(G\) is the smallest integer \(k\) such that \(G \cup N_k\) is vertex-Euclidean. To illustrate the idea behind this new graph labeling problem, we study the vertex-Euclidean deficiency of two new families of graphs called the complete fan graphs and the complete wheel graphs. We also explore some related problems, and pose several research topics for further study.

A signed magic rectangle \(SMR(m, n; r, s)\) is an \(m \times n\) array with entries from \(X\), where \(X = \{0, \pm1, \pm2, \ldots, \pm(ms – 1)/2\}\) if \(mr\) is odd and \(X = \{\pm1, \pm2, \ldots, \pm mr/2\}\) if \(mr\) is even, such that precisely \(r\) cells in every row and \(s\) cells in every column are filled, every integer from set \(X\) appears exactly once in the array, and the sum of each row and of each column is zero. In this paper, we prove that a signed magic rectangle \(SMR(m, n; r, 2)\) exists if and only if \(m = 2\) and \(n = r \equiv 0, 3 \pmod{4}\) or \(m, r \geq 3\) and \(mr = 2n\).

A graph \(G\) is asymmetric if its automorphism group is trivial. Asymmetric graphs were introduced by Erdős and Rényi [1]. They suggested the problem of starting with an asymmetric graph and removing some number, \(r\), of edges and/or adding some number, \(s\), of edges so that the resulting graph is non-asymmetric. Erdős and Rényi defined the degree of asymmetry of a graph to be the minimum value of \(r+s\). In this paper, we consider another property that measures how close a given non-asymmetric graph is to being asymmetric. Brewer et al defined the asymmetric index of a graph \(G\), denoted \(ai(G)\) is the minimum of \(r+s\) so that the resulting graph \(G\) is asymmetric [2]. It is noted that \(ai(G)\) is only defined for graphs with at least six vertices. We investigate the asymmetric index of both connected and disconnected graphs including paths, cycles, and grids, with the addition of up to two isolated vertices. Furthermore for a graph in these families \(G\) we determine the number of labelled asymmetric graphs that can be obtained by adding or removing \(ai(G)\) edges. This leads to the related question: Given a graph \(G\) where \(ai(G)=1\), what is the probability that for a randomly chosen edge \(e\), that \(G-e\) will be asymmetric? A graph is called minimally non-asymmetric if this probability is \(1\). We give a construction of infinite families of minimally non-asymmetric graphs.

A graph \(G\) is said to arrow the graphs \(F\) and \(H\), written \(G \rightarrow (F, H)\), if every red-blue coloring of \(G\) results in a red \(F\) or a blue \(H\). The primary question has been determining graphs \(G\) for which \(G \rightarrow (F, H)\). If we consider the version for which \(F = H\), then we can ask a different question: Given a graph \(G\), can we determine all graphs \(F\) such that \(G \rightarrow (F, F)\), or simply \(G \rightarrow F\)? We call this set of graphs the down-arrow Ramsey set of \(G\), or \(\downarrow G\). The down-arrow Ramsey set of cycles, paths, and small complete graphs are determined. Graph ideals and graph intersections are introduced and a method for computing down-arrow Ramsey sets is described.

We use a dynamic programming algorithm to establish a new lower bound on the domination number of complete cylindrical grid graphs of the form \(C_n\square P_m\), that is, the Cartesian product of a path and a cycle, when \(n\equiv 2\pmod{5}\), and we establish a new upper bound equal to the lower bound, thus computing the exact domination number for these graphs.

In this paper, we address computational questions surrounding the enumeration of non-isomorphic André planes for any prime power order \(q\). We are particularly focused on providing a complete enumeration of all such planes for relatively small orders (up to 125), as well as developing computationally efficient ways to count the number of isomorphism classes for other orders where enumeration is infeasible. André planes of all dimensions over their kernel are considered.

We use a dynamic programming algorithm to establish a lower bound on the domination number of complete grid graphs of the form \(C_n\square P_m\), that is, the Cartesian product of a cycle \(C_n\) and a path \(P_m\), for \(m\) and \(n\) sufficiently large.

With the increasingly frequent exchanges between countries, my country’s demand for high-quality English translators has greatly increased. However, an important problem we are currently facing is that China’s translation talents are far behind the demand. An important reason for this phenomenon is that the traditional translation teaching is difficult to cultivate translators who can meet the market demand. Therefore, it is necessary to improve the traditional translation teaching mode. Translation teaching for English majors is an important part of translation teaching. Therefore, after evaluating the speech characteristics and speech data, this document first proposes a translation classification error detection model based on mfcc-rf. The acoustic function of the extracted 39 dimensional Mel inverse spectral coefficient is the input of the random forest classifier, and a classification error detection model is established. By analyzing the experimental results, the MFCC radio frequency translation error detection model has achieved high classification error detection accuracy under three types of errors (rising, falling and shortening). The experimental results show that, with semantic similarity as the design principle of distractors, using the word vector training method of the context word prediction model to automatically generate distractors can ultimately improve the comprehensive training efficiency of college English majors’ translation ability.

A node in the \(n\)-dimensional hypercube \(Q_n\) is called an odd node (resp. an even node) if the sum of all digits of the node is odd (resp. even). Let \(F\subset E(Q_n)\) and let \(L\) be a linear forest in \(Q_n-F\) such that \(|E(L)|+|F|\leq n-2\) for \(n\geq 2\). Let \(x\) be an odd node and \(y\) an even node in \(Q_n\) such that none of the paths in \(L\) has \(x\) or \(y\) as internal node or both of them as end nodes. In this note, we prove that there is a Hamiltonian path between \(x\) and \(y\) passing through \(L\) in \(Q_n-F\). The upper bound \(n-2\) on \(|E(L)|+|F|\) is sharp.

As a product of the revolutionary war years, red culture, with its strong vitality, strong cohesion and extraordinary charm, with its incomparable positive energy, resists vulgar and flattering culture, promotes people to rebuild their faith, purify their minds, stimulate their motivation, and promote the process of cultural power. Yan’an, represented by red culture, is rich in resources. This is the holy land of Chinese revolution, the first batch of famous historical and cultural cities named by the State Council, and the three major education bases of patriotism, revolutionary tradition, and Yan’an spirit. The development and utilization of such resources have great political, cultural, educational and economic values. This research is based on the development of red culture, and uses the distributed machine learning system to realize in the system architecture of parameter server. In the distributed system set in this study, node downtime and network interruption are random. When the parameter server system adopts static scheduling, it leads to poor scalability and robustness. The experimental results show that under the intelligent simulation of machine learning system, the development of red culture resources meets the expected assumptions, and the accuracy of the model is relatively high.

In this paper, we introduce a class of restricted symmetric permutations, called half-exceeded symmetric permutations. We deduce the enumerative formula of the permutations of \(\{1,2,\ldots,2n\}\) and give it a refinement according to the distribution of the inverse pairs. As a consequence, we obtain new combinatorial interpretations of some well-known sequences, such as Stirling numbers of the second kind and ordered Bell numbers. Moreover, we introduce the ordered Stirling number of the second kind and establish a combinatorial proof of the recursive relation of the sequence.

The total labeling of a graph \(G=(V,E)\) is a bijection from the union of the vertex set and the edge set of \(G\) to the set \(\{1,2,\dots,|V(G)|+|E(G)|\}\). The edge-weight of an edge under a total labeling is the sum of the label of the edge and the labels of the end vertices of that edge. The vertex-weight of a vertex under a total labeling is the sum of the label of the vertex and the labels of all the edges incident with that vertex. A total labeling is called edge-magic or vertex-magic when all the edge-weights or all the vertex-weights are the same, respectively. When all the edge-weights or all the vertex-weights are different then a total labeling is called edge-antimagic or vertex-antimagic total, respectively.

In this paper we deal with the problem of finding a~total labeling of some classes of graphs that is simultaneously vertex-magic and edge-antimagic or simultaneously vertex-antimagic and edge-magic, respectively.
We show several results for stars, paths and cycles.

This study presents a pioneering federated multi-modal data classification model tailored for smart optical cable monitoring systems. By harnessing federated learning techniques, the model ensures data privacy while achieving performance on par with centralized models. Through comprehensive experiments spanning various modalities, including vision and auditory data, our approach showcases promising outcomes, as evidenced by accuracy and precision metrics. Furthermore, comparative analyses with centralized models highlight the superior data security and reduced network strain offered by federated learning. Moreover, we delineate the design and deployment of a smart optical cable monitoring system leveraging edge computing, accentuating the pivotal role of information technology in elevating operational efficiency within the cable monitoring domain. Through meticulous analysis and simulations, our proposed system adeptly monitors environmental variables, thereby bolstering safety and efficiency in smart optical cable monitoring applications.

The created public art sculpture is a material form that expresses the public spirit of the city. This paper proposes a deep model capable of enhancing the aesthetic quality of public art sculptures. The model uses the inverse mapping network of the augmented network to weaken the restriction of paired data sets required for training, and at the same time designs an effective loss function, that is, constructs the color and texture losses that are actively learned in training through generative adversarial rules, and enhances generative sculpture. The total variational loss of smoothness that improves the aesthetic quality of the sculpture to some extent. On this basis, this paper improves the design idea of content consistency loss. Experiments on the interaction between public art sculptures and the urban environment and the enhancement of aesthetics.

With the increasing scale of college enrollment and the increasing complexity of college teaching management, college finance department should innovate the traditional financial management mode while adapting to the reform of teaching management, and make use of the openness and real-time characteristics of Internet to improve the quality of college financial management and reduce the risk of college financial management. To this end, this paper designs a university financial system based on multi-scale deep learning. In the hardware design, the system adds multiple sensors and scans all the information in the financial database using a coordinator. In the software design, the weights that can connect the financial information of the same attribute are set by establishing a database form; according to the multilayer perceptual network topology, a full interconnection model based on multi-scale deep learning is designed to realize the system’s deep extraction of data. The experimental results show that the financial risk is based on the risk warning capability for university finance, and compared with the system under the traditional design, the university finance system designed in this time has the most categories of financial information parameters extracted.

This work suggests predicting student performance using a Gaussian process model classification in order to address the issue that the prediction approach is too complex and the data set involved is too huge in the process of predicting students’ performance. In order to prevent overfitting, a sample set consisting of the three typical test outcomes from 465 undergraduate College English students is divided into training and test sets. The cross-validation technique is used in this study. According to the findings, Gaussian process model classification can accurately predict 92% of the test set with a prediction model, and it can also forecast students’ final exam marks based on their typical quiz scores. Furthermore, it is discovered that the prediction accuracy increases with the sample set’s distance from the normal distribution; this prediction accuracy rises to 96% when test scores with less than 60 points are taken out of the analysis.

Fix integers \(k, b, q\) with \(k \ge 2\), \(b \ge 0\), \(q \ge 2\). Define the function \(p\) to be: \(p(x) = kx + b\). We call a set \(S\) of integers \emph{\((k, b, q)\)-linear-free} if \(x \in S\) implies \(p^i(x) \notin S\) for all \(i = 1, 2, \dots, q-1\), where \(p^i(x) = p(p^{i-1}(x))\) and \(p^0(x) = x\). Such a set \(S\) is maximal in \([n] := \{1, 2, \dots, n\}\) if \(S \cup \{t\}, \forall t \in [n] \setminus S\) is not \((k, b, q)\)-linear-free. Let \(M_{k, b, q}(n)\) be the set of all maximal \((k, b, q)\)-linear-free subsets of \([n]\), and define \(g_{k, b, q}(n) = \min_{S \in M_{k, b, q}(n)} |S|\) and \(f_{k, b, q}(n) = \max_{S \in M_{k, b, q}(n)} |S|\). In this paper, formulae for \(g_{k, b, q}(n)\) and \(f_{k, b, q}(n)\) are proposed. Also, it is proven that there is at least one maximal \((k, b, q)\)-linear-free subset of \([n]\) with exactly \(x\) elements for any integer \(x\) between \(g_{k, b, q}(n)\) and \(f_{k, b, q}(n)\), inclusively.

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.