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.