Optimizing hybrid network topologies in communication networks through irregularity strength

Graph theory has emerged as an influential tool for communication network design and analysis, especially for designing hybrid network topologies for local area networks (LANs). LAN topologies often face challenges related to scalability, data traffic optimization, and security. Designing reliable and efficient hybrid LAN structures remains a critical problem in communication networks. This paper addresses the issue by proposing the application of graph labeling techniques, particularly H-irregularity strength, as a mathematical framework to model and optimize hybrid network topologies. The results illustrate the way theoretical graph labeling and practical network technology interact, offering a novel solution to LAN design problems. This study adds to the expanding field of graph theory applications in communication networks by relating graph theoretical ideas to actual network topologies. With an emphasis on the irregularity strength of particular graph families, the role of graph labeling in optimizing these topologies is explored in this study. The labeling methods that are given offer valuable insights into improving communication efficiency, guaranteeing LAN scalability, and optimizing network architecture. The theoretical underpinnings of the application of graph theory to communication network modelling are strengthened by these discoveries. Labeling methods are introduced in this study to capture topological irregularities and labeling constraints through the use of specialized graphs, such as the Dutch Windmill and Corona product graphs as they both have special labeling characteristics that can be used to improve network performance. In order to secure data and prevent network failures, a three-unit organization structure with a shared administrator is used in network design and optimization. A model of hybrid ring topology of a local area network is considered in this paper and different models are presented which are originated from Dutch Windmill and corona product of graphs. The contribution of this paper is it includes results about a special version of irregularity strength in which the subgraphs used are Dutch windmill graphs and cycle graphs. The edge, vertex and total H-irregularity strength of Dutch Windmill graph and Corona product graph are calculated, offering fresh perspectives on the way they could represent hybrid LAN topologies. The irregularity strength metric is particularly useful even though it measures the imbalance in vertex degrees, which is essential for optimizing communication flow and load balancing within a network. Our theoretical findings illustrate how these labeling schemes can model network behavior, improve resource allocation, and trace data flow effectively. Although the study is primarily theoretical, it offers groundwork for practical network simulation and real-world implementation. Future work will focus on validating the models through simulations and assessing performance metrics such as latency, throughput, and fault recovery. A key limitation of the current study is the absence of empirical performance validation, which is identified as an important direction for further research.