On Heronian mean anti-magic labeling of some graphs

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) = ∑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.