Purwanto, Bait Imala
Let G be a graph having vertex set V(G), edge set E(G), |V(G)| = p, |E(G)| = q, and A be { 0,1,2,⋯, [q2] }. A vertex equitable labeling of G is a labeling f: V(G) → A that induces a bijective labeling of edges f*: E(G) → {1,2, ⋯, q}, where f*(uv) = f(u) + f(v) for every uv ∈ E(G), such that |vf(a) - vf (b)| ≤ 1, Aa,b ∈ A, where vf(a) be the number of vertices v with f(v) = a for a ∈ A. A graph G is vertex equitable if there exists a vertex equitable labelling of G. Many authors have studied vertex equitable labeling, and they found many vertex equitable graphs. Many graphs are not known whether they are vertex equitable or not. We need to find some new classes of graphs that are vertex equitable. Let m and n be positive integers. An actinia graph A(m, n) is a unicyclic graph obtained from a cycle Cm, m ≥3, and each vertex of Cm is joined to n new vertices of null graph Nn. In this article we find that actinia graph A(m, n) is vertex equitable for every even m. © 2022 Author(s).
Department of Mathematics, Universitas Negeri Malang, Jalan Semarang 5, East Java, Malang, 65145, Indonesia