Further Results on (a, d) -total Edge Irregularity Strength of Graphs

Abstract

Consider a simple graph 𝐺 = (𝑉, 𝐸) on 𝑙 vertices and 𝑚 edges together with a total ℎ – labeling 𝜌: 𝑉(𝐺) ∪ 𝐸(𝐺) → {1,2,3, … , ℎ}. Then ρ is called (𝑎, 𝑑)–total edge irregular labeling if there exists a one-to-one correspondence, say 𝜓: 𝐸(𝐺) → {𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, … + 𝑎 + (𝑚 − 1)𝑑} defined by 𝜓(𝑢𝑣) = 𝜌(𝑢) + 𝜌(𝑣) + 𝜌(𝑢𝑣) for all 𝑢𝑣 ∈ 𝐸(𝐺), where 𝑎 ≥ 3, 𝑑 ≥ 2. Also, the value 𝜓(𝑢𝑣) is said to be the edge weight of 𝑢𝑣. The (𝑎, 𝑑) −total edge irregularity strength of the graph G is indicated by (𝑎, 𝑑) − 𝑡𝑒𝑠(𝐺) and is the least ℎ for which G admits (𝑎, 𝑑) – edge irregular h-labeling. In this article, (𝑎, 𝑑) − 𝑡𝑒𝑠(𝐺) for some common graph families are examined. In addition, an open problem (3,2)– 𝑡𝑒𝑠(𝐾_(𝑚, 𝑛) ), 𝑚, 𝑛 > 2 is solved affirmatively.