Innovative Lucky-edge Odd Mean Labeling of Star-Related Graphs: Bridging Theory and Circuit Design

Abstract

This study explores the application of Lucky-edge Odd Mean Labeling (LEOML) to special classes of star-related graphs, including Middle, Total, and Central graphs. LEOML is an exclusive edge labeling technique in which the mean of edge labels incident to each vertex is controlled to be an odd number. The goal is to tie theoretical graph theory with practical applications in network and circuit design, providing intuitions into the structural and functional rewards of this labeling approach. We explore the theoretical implications of LEOML by computing the Lucky-edge Odd Mean Number (LEOMLN) for various special classes of star-related graphs. Furthermore, we examine the relationship between LEOMLN and the graph energy of circuit model representations. The energy of a graph, derived from its adjacency matrix and eigenvalues, offers intuitions into the graph's spectral properties, which are essential for understanding its performance in practical applications. The results reveal a significant correlation between LEOMLN and graph energy, opening up new opportunities for optimizing network and circuit performance based on theoretical graph properties. This study assists as a bridge between abstract graph theory and real-world applications, providing valuable knowledge to both theoretical research and practical design disciplines. The innovative application of LEOML has the potential to impact various fields, including network optimization, circuit design, and computational analysis.