Investigation of Compact Mean Labeling Using Complete Bipartite Graph

Assigning numbers to a graph's vertices in a way
that ensures the mean value of labels at nearby vertices meets
predetermined criteria—often connected to edge labels or other
structural constraints—is known as compact mean labelling. In
the setting of complete bipartite graphs, a basic type of graphs in
which vertices are separated into two disjoint sets and each vertex
in one set is related to all vertices in the other, this study
investigates compact mean labelling. In full bipartite graphs, we
examine the requisite and adequate requirements for the presence
of compact mean labelling. The paper presents labelling
techniques that guarantee adherence to the mean labelling
requirements while minimising label range. Such labelings have
applications in combinatorial designs, network topology, and
coding theory, where effective and structured labelling is essential
for maximising data organisation and efficiency.