Advanced computational methods for compact mean labeling in star graphs

Graph labeling involves assigning unique integer values to the vertices and edges of a graph, starting from zero or a higher value. A star graph, denoted as K (1, r), consists of a central vertex connected to r peripheral vertices, forming a complete bipartite graph. In this research, the notion of compact mean labeling using star graphs is explored. Typically, a star graph has p< q edges. However, for compact mean labeling, the graph should have p≤ q. To resolve this inconsistency, the vertex labeling of the graph has been determined from zero to 3q-p. This proposed extended compact mean labeling technique is verified for the bi-star graphs, the join sum of the star graph, and the union of the star graph. Our findings offer new insights into the characteristics and application of star graphs in extended compact mean labeling, particularly in comparing the strong and weak ties within a network.