Weaker and deficiency of even vertex odd edge root square mean labeling graphs

This paper introduces and investigates a relaxed variant of even vertex odd edge root square mean labeling (EVOERSML), called weaker EVOERSML. For a graph G, a weaker EVOERSML is an injective labeling f : V(G) → {0, 2, 4, . . . , 2(q + k)}, where k ∈ Z + , such that all vertex labels
are distinct non-negative even integers. The induced edge labels are obtained by applying either the floor or ceiling function to the square root of the average of the squares of the labels of the end vertices. This relaxed labeling framework extends the applicability of root square mean labelings to a broader class of graphs. The minimum labeling bound Kmin(G) is defined as the least integer k for which G admits a weaker EVOERSML, and methods for determining this bound are discussed. The existence of weaker EVOERSML is established for several families of graphs, and their
structural properties are analyzed. The deficiency associated with weaker EVOERSML is also examined, with particular emphasis on connected graphs of orders 3, 4, and 5, yielding complete classifications and illustrative examples.