Journal of Prime Research in Mathematics

Characterizing Irregularity in Planar Graph Structures

Abdul Aleem Mughal, Raja Noshad Jamil, Abaid ur Rehman Virk \(^{∗,∗}\)

\(^a\)University of Management and Technology, Lahore, Pakistan.

\(^b\)University of Management and Technology, Lahore, Pakistan


Correspondence should be addressed to:,,


Face irregularity strength under ρ−labeling ξ with class (α1, β1, γ1) of plane graphs is a labeling from the set of graph elements into the set of integers, that is, ξ: {V ∪ E ∪ F} → {1, 2, 3, .., ρ}, such that the face weights are distinct at any stage in the graph labeling, that is, Wξ(α1,β1,γ1)(f) ̸= Wξ(α1,β1,γ1)(g), for any two faces f and g of the graph G. The face irregular strength of a plane graph G is the least possible integer ρ such that G admits face irregular ρ−labeling. In this research, authors have examined the exact tight lower bounds for the face irregular strength of generalized plane graphs under ρ−labeling of class (α1, β1, γ1) for vertex (1, 0, 0), edge (0, 1, 0), face (0, 0, 1), vertex-face (1, 0, 1), edge-face (0, 1, 1) and entire (1, 1, 1). Results are verified by examples.


Cartesian product, Face labeling, Face irregularity strength, Face weights.