Even vertex odd edge root square mean labeling of some cycle-related graphs

Abstract

An injective mapping (f:V(G)\rightarrow \{0,2,4,...,2|E|\}\) is said to satisfy the even vertex odd edge root square mean labeling (EVOERSML) condition when vertices are assigned distinct even values and each edge \(uv\in E(G)\) receives a distinct odd label determined by \( \left\lceil \sqrt{\frac{f(u)²+f(v)²}\{2}} \right\rceil \quad or \quad \left\lfloor \sqrt{\frac{f(u)²+f(v)²}\{2}} \right\rfloor.\) Graphs that admit such assignments are called EVOERSML graphs. The present study examines this labeling scheme on several graph structures closely related to cycles. Constructive methods are provided for families including ladder graphs, tadpole graphs, polygon-chain graphs, dumbbell graphs, and sunlet graphs. The obtained results emphasize how parity restrictions influence labeling existence and uniqueness. This work enriches the theory of root square mean labeling and opens further scope for studying EVOERSML-type labelings in more complex graph families and applied network models.