ON HERONIAN MEAN ANTI-MAGIC LABELING OF SOME GRAPHS

ON HERONIAN MEAN ANTI-MAGIC LABELING OF SOME GRAPHS

B. Sivaranjani, R. Kala

[PDF]

Abstract

Let G = (V (G),E(G)) be a finite, simple, connected and undirected graph with p vertices and q edges. Let Φ : V (G) → {1, 2, 3, · · · , q+1} and the induced edge labeling Φ ∗ : E(G) → {1, 2, 3, · · · , q+1} is defined by Φ ∗ (e = uv) = ⌈ Φ(u)+ √ Φ(u)Φ(v)+Φ(v) 3 ⌉ or ⌊ Φ(u)+ √ Φ(u)Φ(v)+Φ(v) 3 ⌋ for e ∈ E(G). Then Φ is said to be a Heronian mean labeling if induced edge labels Φ ∗ (e) are distinct. An anti-magic labeling is a bijection from the set of edges to the set of integers {1, 2, 3, · · · , q} such that the weights are pairwise distinct, where the weight at one vertex is the sum of all labels of the edges incident to such vertex. A Heronian mean labeling Φ is said to be Heronian mean anti-magic if w(vi) ̸= w(vj) for all distinct vertices vi, vj ∈ V (G), where w(v) = Σ u∈N(v) Φ ∗ (uv). A graph is called Heronian mean anti-magic graph if it admits a Heronian mean anti-magic labeling. In this paper, we investigate the behaviour of this labeling for graphs which contains a clique of order at least 4, Pn ◦ 2K1, kCn, n ≥ 4, CLn, n ≥ 3, T ∪ T ′ where T and T ′ be any two trees of order at least 3. We also prove that K2,n is Heronian mean anti-magic for n ≤ 9 and is not for n ≥ 10.

Keywords

Mean labeling, Anti-magic labeling, Mean anti-magic labeling, Heronian mean anti-magic labeling.