The multiplicative sum Zagreb index is a modified version of the well-known Zagreb indices. The multiplicative sum Zagreb index of a graph \(G\) is the product of the sums of the degrees of pairs of adjacent vertices. The mathematical properties of the multiplicative sum Zagreb index of graphs with given graph parameters deserve further study, as they can be used to detect chemical compounds and study network structures in mathematical chemistry. Therefore, in this paper, the maximal and minimal values of the multiplicative sum Zagreb indices of graphs with a given clique number are presented. Furthermore, the corresponding extremal graphs are characterized.

Let \( G = (V, E) \) be a graph. A subset \( S \subseteq V \) of vertices is an \textit{efficient dominating set} if every vertex \( v \in V \) is adjacent to exactly one vertex in \( S \), where a vertex \( u \in S \) is considered to be adjacent to itself. Efficient domination is highly desirable in many real-world applications, and yet, in general, graphs are often not efficient. It is of value, therefore, to determine optimum ways in which inefficient graphs can be changed in order to make them efficient. It is well known, for example, that almost no \( m \times n \) grid graphs have efficient dominating sets. In this paper, we consider the minimum number of vertices that can be removed from an \( m \times n \) grid graph so that the remaining graph has an efficient dominating set.

Let \( G = (V, E) \) be any graph. If there exists an injection \( f : V \rightarrow \mathbb{Z} \), such that \( |f(u) – f(v)| \) is prime for every \( uv \in E \), then we say \( G \) is a prime distance graph (PDG). The problem of characterizing the family of all prime distance graphs (PDGs) with chromatic number 3 or 4 is challenging. In the fourth part of this series of articles, we determined which fans are PDGs and which wheels are PDGs. In addition, we showed: (1) a chain of \( n \) mutually isomorphic PDGs is a PDG, and (2) the Cartesian product of a PDG and a path is a PDG. In this part of the series, we improve (1) by showing that there exists a chain of \( n \) arbitrary PDGs which is a PDG. We also show that the following graphs are PDGs: (a) any graph with at most three cycles, (b) the one-point union of cycles, and (c) a family of graphs consisting of paths with common end vertices.

Let \(P_n\) and \(K_n\) respectively denote a path and complete graph on \(n\) vertices. By a \(\{pH_{1}, qH_{2}\}\)-decomposition of a graph \(G\), we mean a decomposition of \(G\) into \(p\) copies of \(H_{1}\) and \(q\) copies of \(H_{2}\) for any admissible pair of nonnegative integers \(p\) and \(q\), where \(H_{1}\) and \(H_{2}\) are subgraphs of \(G\). In this paper, we show that for any admissible pair of nonnegative integers \(p\) and \(q\), and positive integer \(n \geq 4\), there exists a \(\{pP_{4}, qS_{4}\}\)-decomposition of \(K_n\) if and only if \(3p+4q=\binom{n}{2}\), where \(S_4\) is a star with \(4\) edges.

Natural environment protection compensation refers to the legal system that protects the natural ecological environment, protects the natural environment and makes the beneficiaries of the natural environment get compensation by some means, so as to adjust the interests of the relevant subjects of natural ecological environment protection. This paper discusses the ecosystem service function and its type division of newly built areas in Ganjiang, and the emergy evaluation of ecosystem service function of newly built areas in Ganjiang, establishes the regular scheduling and joint optimal scheduling models of natural ecosystem service value single reservoir, intro-duces the corresponding model solving methods, and applies the ant colony algorithm to the optimal schedule is a lesson. According to the ant colony algorithm, the best way to study the region is to determine these algorithms. Combined with the kernel density analysis method, the spatial scope, potential corridors and key recovery points of ecological corridors are identified, and the optimization mode of natural ecological security pattern of Shule River is constructed. The experimental results show that the optimized ant colony algorithm proves that joint scheduling plays a more prominent role in ecological environment protection, mainly in ecological support and ecological regulation. At the same time, it verifies the applicability of ant colony algorithm in joint scheduling, and improves the average protection efficiency of natural ecology to 20.9%.

The semantic function of modern Chinese “negation + X” modal words based on communication technology and big data corpus has gained wide attention. As the basis of SOA architecture, Web services provide the key resources for worldwide information transfer and information sharing with their characteristics of loose coupling, platform independence, and data exchange without additional support from third-party hardware and software. However, along with the popularization and improvement of Web service technology, the number and types of Web services in the Internet are also increasing massively, and there are a large number of Web services with various functions, quality and granularity. Therefore, how to quickly and accurately discover Web services that satisfy users’ query requests from a large and complex set of services has become a critical problem to be solved in the current Web service discovery research. Based on the real corpus, this paper analyzes the similarities and differences in the semantic functions of modern Chinese “negation + X” modal words by combining lexicalization and grammatization, cognitive linguistics, systemic functional grammar and other related theories. The experimental results demonstrate that the model is designed for automatic annotation of semantic word classes, and the annotation algorithm based on the hidden horse model, combined with the Viterbi algorithm based on dynamic programming, achieves a correct rate of 94.3% in the closed test and 89.1% in the open test despite the small size of the training corpus and severe data sparsity, and the model fitting effect meets the dynamic expectations.

The relationship between competition state anxiety, motor motivation and coping styles of adolescent track and field athletes in China was investigated using interview and questionnaire research methods. The results showed that the mean scores of cognitive state anxiety and somatic state anxiety were lower in junior track and field athletes who had entered the echelon for a short period of time than in older athletes, and the opposite was true for state self-confidence; there were highly significant differences and significant differences in the identity regulation and introjection regulation dimensions of motor motivation; and there were significant differences in the focused problem-solving coping dimension of coping style. This paper proposes an algorithm for classifying athletic visual mirrors based on sequential model mining. This paper focuses on two issues – feature extraction and definition of semantic rules. In the feature extraction stage, the track and field video footage is automatically segmented into a series of identifiable sequences of athletic events, and then each type of behavioral event is identified using a mechanically learned algorithm. There were no significant differences between the three age groups in terms of race state anxiety, identity regulation and introjection regulation, and no significant differences in coping styles. There were no significant differences in the anxiety of competition status, motivation and coping styles among youth athletes of different sport levels. The results showed the effectiveness of the present algorithm for classifying track and field video cameras.

A graph labeling is an assignment of integers to the vertices or edges or both, which satisfies certain conditions. The domination cover pebbling number of a graph \( G \) is \( \psi(G) \), which is the minimum number of pebbles required such that any initial configuration of \( \psi(G) \) pebbles can be transformed through a number of pebbling moves so that the set of vertices with pebbles after the pebbling operation forms a dominating set of \( G \). In this paper, we explore the relationship between two graph parameters, namely graph labeling and domination cover pebbling.

In this paper, we study the \( A_\alpha \)-spectral radius of graphs in terms of given size \( m \) and minimum degree \( \delta \geq 2 \), and characterize the corresponding extremal graphs completely. Furthermore, we characterize extremal graphs having maximum \( A_\alpha \)-spectral radius among (minimally) \( 2 \)-edge-connected graphs with given size \( m \).

The metric dimension of a graph \(\Gamma = (V, E)\), denoted by \( \operatorname{dim}(\Gamma) \), is the least cardinality of a set of vertices in \(\Gamma\) such that each vertex in \(\Gamma\) is determined uniquely by its vector of distances to the vertices of the chosen set. The topological distance between an edge \(\varepsilon = yz \in E\) and a vertex \( k \in V \) is defined as \( d(\varepsilon, k) = \min\{d(z, k), d(y, k)\} \). A subset of vertices \( R_{\Gamma} \) in \( V \) is called an edge resolving set for \(\Gamma\) if for each pair of different edges \( e_{1} \) and \( e_{2} \) in \( E \), there is a vertex \( j \in R_{\Gamma} \) such that \( d(e_{1}, j) \neq d(e_{2}, j) \). An edge resolving set with minimum cardinality is called the edge metric basis for \(\Gamma\) and this cardinality is the edge metric dimension of \(\Gamma\), denoted by \( \operatorname{dim}_{E}(\Gamma) \). In this article, we show that the cardinality of the minimum edge resolving set is three or four for two classes of convex polytopes (\( S_{n} \) and \( T_{n} \)) that exist in the literature.