Farey Graceful Labeling of Some Complex Structures

Graph labeling is a significant area of graph theory due to its wide range of theoretical and practical applications. Recently, Kumar et al. introduced the concept of Farey graceful labeling based on the Farey sequence, which consists of reduced rational numbers between and arranged in increasing order. In Farey graceful labeling, vertices of a graph are assigned distinct terms of the Farey sequence such that the induced edge labels satisfy prescribed conditions. This labeling scheme was shown to exist for several classes of graphs, including paths, cycles, hairy cycles, caterpillar graphs and super caterpillar graph. In this paper, we extend the study of Farey graceful labeling to additional families of graphs. Specifically, we prove that the graphs, double wheel graphs , graph obtained by switching of vertices of cycle, admit Farey graceful labeling. The results presented in this work contribute to the growing literature on rational graph labeling and open new directions for further investigation of Farey-based labeling techniques.