Investigation of super lehmer – 3 mean labeling in theta related graphs

In graph theory, labeling of vertices and edges according to specific rules offers insight into the structural properties of graphs and their potential applications. This study focuses on a particular type of graph labeling known as Super Lehmer-3 Mean Labeling. A graph is said to admit such a labeling if there exists an injective vertex labeling such that the corresponding edge labels, derived from the Lehmer-3 mean of their incident vertex labels, are also distinct and satisfy certain arithmetic conditions. Graphs that admit such a labeling are termed Super Lehmer-3 Mean Graphs.In this work, we investigate the existence of super Lehmer-3 mean labeling in various families of theta-related graphs. Specifically, we analyze standard theta graphs and their structural variations obtained by subdivisions and attachments of pendant vertices. The results confirm that these modified forms of theta graphs also admit super Lehmer-3 mean labeling. Additionally, we extend our exploration to harmonic mean labeling in selected theta graphs, highlighting the conditions under which this alternative labeling scheme is valid.The findings presented in this paper contribute to the growing body of knowledge in graph labeling theory, offering new characterizations and constructions of labeled graph families. These results may have implications for theoretical studies and applications where such labeling structures are relevant.