On the Index Set of Complete Multipartite Graphs

Document Type : Original Research Article

Author

Semnan Branch, Islamic Azad University, Semnan, Iran

Abstract

For every natural number $h$, a graph $G$ is said to be $h$-magic if there exists a labeling $l:E(G)\longrightarrow {Z}_h\setminus \{ 0 \}$ such that the labeling $l^+:V(G)\longrightarrow Z_{h}$ defined by $$l^+(v)=\sum_{uv\in E(G)} l(uv), $$ is a constant map. A constant of a magic sum is called an index of $G$, an index for short, and we write ${\rm I}_{A}(G)=\{r: \hspace{.1cm} G \hbox{ is } A\hbox{-magic with index } r\}$. Let $G$ be a graph and $A$ be an abelian group. A graph $G$ is called $A$-magic if there exists an edge labeling $l:E(G)\longrightarrow A\setminus \{ 0 \}$ such that the induced vertex set labeling $l^+:V(G)\longrightarrow A$, defined by $l^+(v)=\sum_{uv\in E(G)} l(uv)$, where the sum is over all $e\in E(G)$ incident with $v$, is a constant map. A constant of a magic sum is called an index of $G$, an index for short, and we write ${\rm I}_{A}(G)=\{r: \hspace{.1cm} G \hbox{ is } A\hbox{-magic with index } r\}$. In 2011 Shiu and Low proved that $0\in {\rm I}_{A}(K_{n_{1},\ldots,n_{t}})$, where $n_{i}\geq2$ ($i=1,\ldots,t$). For an undirected graph G, and an abelian group A, an A-magic labelling is an assignment of non-zero element of A, to the edges of G, such that the sum of the values of all edges incident with each vertex is constant. A constant on magic sum is called an index set of G. In this paper, for $t\geq2$ we determine the index set of the complete multipartite graph $K_{n_{1},\ldots,n_{t}}$, where $n_{i}\geq2$ (for $i=1,\ldots,t$).

Keywords

References

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 8, Issue 2
December 2023
Pages 153-159

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History

  • Receive Date: 14 January 2025
  • Revise Date: 12 February 2025
  • Accept Date: 17 February 2025
  • Publish Date: 01 December 2024

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