Abstract
Let be an abelian group under multiplication. Let . Then the vertex magic labeling on is induced as such that where the product is taken over all edges of incident at is constant. A graph is said to be - magic if it admits a vertex magic labeling on . In this paper, we prove that , , Generalized fish graph, Double cone graph and four Leaf Clover graph are all -magic graphs.
Highlights
For a non-trivial abelian group V4 under multiplication a graph G is said to be V4 -magic graph if there exist a labeling g of the edges of G with non-zero elements of V4 such that the vertex labeling g∗ defined as g∗(v) = ∏u g(uv) taken over all edges uv incident at v is a constant.Let V4 = {i, −i, 1, −1} we have proved that the Cartesian product of two graphs,Generalized fish graph, Happy graph,Four Leaf Clover Graph are all V4 -magic graphs.2
Basic Definition Definition: 2.1Cartesian Product of Two graphs Cartesian product of two graphs G, H is a new graph GH with the vertex set V × V and two vertices are adjacent in the new graph if and only if either u = vand u′ is adjacent to v′ in H or u′ = v′ and u is adjacent to v in G
Definition: 2.2Generalized Fish Graph The generalized fish graph is defined as the one point union of any even cycle with C3
Summary
Article History : Received :11 January 2021; Accepted: 27 February 2021; Published online: 5 April 2021 Abstract: Let V4 be an abelian group under multiplication. The vertex magic labeling on V4 is induced as g∗: V(G) → V4 such that g∗(v) = ∏u g(uv) where the product is taken over all edges uv of G incident at v is constant. A graph is said to be V4 - magic if it admits a vertex magic labeling on V4. We prove that Cm × Cn,m ≥ 3, n ≥ 3, Generalized fish graph, Double cone graph and four Leaf Clover graph are all V4 -magic graphs. Keyword: Vertex magic labeling on V4, V4 -magic graph, Four Leaf Clover Graph.
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