Graphs of order n with fault-tolerant metric dimension n have recently been characterized.This paper points out an error in the proof of this characterization. We show that the complete multipartite graphs also have the fault-tolerant metric dimension n, which provides an infinite family of counterexamples to the characterization. Furthermore, we find exact values of the metric, edge metric, mixed-metric dimensions, the domination number, locating-dominating number, and metric-locating-dominating number for the complete multipartite graphs. These results generalize various results in the literature from complete bipartite to complete multipartite graphs.

The characteristics of MANET such as decentralized architecture, dynamic topologies make MANETs susceptible to various security attacks. Sequence number attacks are such type of security threats which tend to degrade the network functioning and performance by sending fabricated route reply packets (RREP) with the objective of getting involved in the route and drop some or all of the data packets during the data transmission phase. The sequence number adversary attempts to send a fabricated high destination number in the RREP packet which attracts the sender to establish a path through the adversary node. This paper proposes a proactive secure routing mechanism which is an improvement over the authors previously proposed scheme. It makes use of linear regression mechanism to predict the maximum destination sequence number that the neighboring node can insert in the RREP packet. As an additional security checkpoint, it uses a bait detection mechanism to establish confidence in marking a suspicious node as a malicious node. The proposed approach works in collaboration with the ad hoc on-demand distance vector routing (AODV) protocol. The simulation results depict that the approach improves the network performance in the presence of adversaries as compared to previously proposed scheme.

Choosing the optimal path in planning is a complex task due to the numerous options and constraints; this is known as the trip design problem (TTDP). This study aims to achieve path optimization through the weighted sum method and multi-criteria decision analysis. Firstly, this paper proposes a weighted sum optimization method using a comprehensive evaluation model to address TTDP, a complex multi-objective optimization problem. The goal of the research is to balance experience, cost, and efficiency by using the Analytic Hierarchy Process (AHP) and Entropy Weight Method (EWM) to assign subjective and objective weights to indicators such as ratings, duration, and costs. These weights are optimized using the Lagrange multiplier method and integrated into the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) model. Additionally, a weighted sum optimization method within the Traveling Salesman Problem (TSP) framework is used to maximize ratings while minimizing costs and distances. Secondly, this study compares seven heuristic algorithms—the genetic algorithm (GA), particle swarm optimization (PSO), the tabu search (TS), genetic-particle swarm optimization (GA-PSO), the gray wolf optimizer (GWO), and ant colony optimization (ACO)—to solve the TOPSIS model, with GA-PSO performing the best. The study then introduces the Lagrange multiplier method to the algorithms, improving the solution quality of all seven heuristic algorithms, with an average solution quality improvement of 112.5% (from 0.16 to 0.34). The PSO algorithm achieves the best solution quality. Based on this, the study introduces a new variant of PSO, namely PSO with Laplace disturbance (PSO-LD), which incorporates a dynamic adaptive Laplace perturbation term to enhance global search capabilities, improving stability and convergence speed. The experimental results show that PSO-LD outperforms the baseline PSO and other algorithms, achieving higher solution quality and faster convergence speed. The Wilcoxon signed-rank test confirms significant statistical differences among the algorithms. This study provides an effective method for experience-oriented path optimization and offers insights into algorithm selection for complex TTDP problems.

Irregularity indices are graph-theoretic parameters designed to quantify the irregularity in a graph. In this paper, we study the practical applicability of irregularity indices in QSPR modeling of the physicochemical and quantum-theoretic properties of compounds. Our comparative testing shows that the recently introduced $IRA$ index has significant priority in applicability over other irregularity indices. In particular, we show that the correlation potential of the $IRA$ index with certain physicochemical and quantum-theoretic properties such as the enthalpy of formation, boiling point, and $\pi$-electron energies is significant. Our QSPR modeling suggests that the regression models with the aforementioned characteristics such as strong curve fitting are, in fact, linear. Considering this the motivation, the $IRA$ index was studied further, and we provide analytically explicit expressions of the $IRA$ index for certain graph operations and compositions. We conclude the paper by reporting the conclusions, implications, limitations, and future scope of the current study.

The quadratic assignment problem (QAP) is a well-known challenging combinational optimization problem that has received many researchers’ attention with varied real-world and industrial applications areas. Using the framework of basic artificial bee colony algorithm, frequently used crossover and mutation operators, and combined with an effective local search method, this paper proposes a simple but effective discrete artificial bee colony (DABC) algorithm for solving quadratic assignment problems (QAPs). Typical QAP benchmark instances are selected from QAPLIB in order to conduct the simulation experiment where common performance metrics are used to evaluate the algorithm. The paper also investigates the influence factors of the algorithm’s performance. The results show that the proposed algorithm is a quite effective and practical new approach for handling QAP problems.