The question on how to colour a graph \( G \) when the number of available colours to colour \( G \) is less than that of the chromatic number \(\chi(G)\), such that the resulting colouring gives a minimum number of edges whose end vertices have the same colour, has been a study of great interest. Such an edge whose end vertices receive the same colour is called a bad edge. In this paper, we use the concept of \(\delta^{(k)}\)-colouring, where \( 1 \leq k \leq \chi(G)-1 \), which is a near proper colouring that permits a single colour class to have adjacency between the vertices in it and restricts every other colour class to be an independent set, to find the minimum number of bad edges obtained from the same for some wheel-related graphs. The minimum number of bad edges obtained from \(\delta^{(k)}\)-colouring of any graph \( G \) is denoted by \( b_k(G) \).

Let \( G \) be a simple finite graph. A \( k \)-coloring of \( G \) is a partition \(\pi = \{ S_1, \cdots, S_k \}\) of \( V(G) \) so that each \( S_i \) is an independent set and any vertex in \( S_i \) takes color \( i \). A \( k \)-coloring \(\pi = \{ S_1, \cdots, S_k \}\) of \( V(G) \) is a neighbor locating coloring if for any two vertices \( u, v \in S_i \), there is a color class \( S_j \) for which one of them has a neighbor in \( S_j \) and the other does not. The minimum \( k \) with this property is said to be the neighbor locating chromatic number of \( G \), denoted by \(\chi_{NL}(G)\). We initiate the study of the neighbor locating coloring of graphs resulting from three types of product of two graphs. We investigate the neighbor locating chromatic number of Cartesian, lexicographic, and corona products of two graphs. Finally, we untangle the neighbor locating chromatic number of any of the aforementioned three products of cycles, paths, and complete graphs.

The significance of advancing socialist S&T legal system building with Chinese features is examined in this paper, along with the extensive effects it has on S&T innovation and economic growth. Against the backdrop of an increasingly competitive global scientific and technology landscape, China’s ability to innovate and achieve sustainable growth hinges on the establishment of an ideal science and technology legislative framework. This essay first examines the primary obstacles to China’s development of a science, technology, and innovation (S&T) legal system, including inadequate protection for intellectual property rights, a flawed process for transforming scientific and technological advancements, and an insufficient system for encouraging enterprise innovation. Then, this research presents a quantitative analysis model to optimize the path of science and technology legal building by applying the improved particle swarm optimization method (PSO). The model takes into account a wide range of variables, including the degree of intellectual property protection, the strength of legal backing, the pace at which scientific and technological advancements are transformed, etc. Through the analysis of simulation data, the model also confirms the promotion effect of the legal system construction on the quantity of patent applications, the success rate of innovation projects, the enterprise R&D expenditure, and the expansion of the local economy. The study’s findings demonstrate that bolstering the science and technology legal system can effectively encourage businesses to boost R&D investment and foster regional economic development in addition to greatly raising the quantity of patent applications and the success rate of innovation projects. The rigorous intellectual property protection laws and ideal legal framework for the conversion of accomplishments greatly boost the regional innovation vitality and economic efficiency, particularly in the case study of Zhongguancun in Beijing and East China. Moreover, adaptive weighting is used to enhance the PSO algorithm and optimize the development of science and technology legal system’s comprehensive performance index, thereby confirming the model’s viability and efficacy. The study’s findings offer theoretical justification and helpful advice for China’s development of a science and technology legal framework, which is crucial for fostering innovation in these fields and boosting the country’s competitiveness.

At present, there are few systematic researches on macro-scale heterogeneous modeling and numerical simulation of dynamic mechanical properties of 3-D braided composites. In this paper, the parametric virtual simulation model of 3D five-directional braided composites is realized in the way of “point-line-solid” based on the integrated design idea of process-structure-performance. And the impact compression numerical simulation of the material is carried out by using multi-scale analysis method. The effects of strain rate and braiding angle on transverse impact compression properties and fracture characteristics of composites is studies and verified by comparing the test results with the numerical simulation results systematically. The dynamic failure mechanism of the matrix and fiber bundles during the impact compression process is revealed. The results show that the macro-scale heterogeneous simulation model of 3D five-directional braided composites established is effective, and the numerical simulation results agree well with the test results. The matrix fracture and shear failure of fiber bundles are presented simultaneously under transverse impact compression. The failure of fiber bundles and matrix mainly concentrates on two main fracture shear planes. And the included angle between the fracture shear planes and the vertical direction is consistent with the corresponding internal braiding angle of specimens.

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

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.

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.