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 \(Bsw(Pa).\)

In this Paper, we establish a new application of the Mittag-Lefier Function method that will enlarge the application to the non linear Riccati Differential equations with fractional order. This method provides results that converge promptly to the exact solution. The description of fractional derivatives is made in the Caputo sense. To emphasize the consistency of the approach, few illustrations are presented to support the outcomes. The outcomes declare that the procedure is very constructive and relavent for determining non linear Ricati differential equations of fractional order.

One of the important features of an interconnection network is its ability to efficiently simulate programs or parallel algorithms written for other architectures. Such a simulation problem can be mathematically formulated as a graph embedding problem. In this paper, we embed complete multipartite graphs into certain trees, such as \(k\)-rooted complete binary trees and \(k\)-rooted sibling trees.

In this paper we compute the \(P_3\)-forcing number of honeycomb network. A dynamic coloring of the vertices of a graph \(G\) starts with an initial subset \(S\) of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set \(S\) is called a forcing set of \(G\) if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set \(S\) has the added property that it induces a subgraph of \(G\) whose components are all paths of length 3, then \(S\) is called a \(P_3\)-forcing set of \(G\). A Ps-forcing set of \(G\) of minimum cardinality is called the \(P_3\)-forcing number of G denoted by \(ZP_3(G)\).

In this paper, we introduced a new concept called nonsplit monophonic set and its relative parameter nonsplit monophonic number \(m_{ns}(G)\). Some certain properties of nonsplit monophonic sets are discussed. The nonsplit monophonic number of standard graphs are investigated. Some existence theorems on nonsplit monophonic number are established.

In graph theory and network analysis, centrality measures identify the most important vertices within a graph. In a connected graph, closeness centrality of a node is a measure of centrality, calculated as the reciprocal of the sum of the lengths of the shortest paths between the node and all other nodes in the graph. In this paper, we compute closeness centrality for a class of neural networks and the sibling trees, classified as a family of interconnection networks.