Line Graph Associated to Graph of a Near-Ring with Respect to an Ideal


  • Moytri Sarmah Department of Mathematicas, Girijananda Chowdhury Institute of Manage- ment and Technology, Guwahati-781017, India
  • Kuntala Patra Department of Mathematics, Gauhati University, Guwahati-781014, India



Graph, Line graph, Near-ring, Ideal


Let $N$ be a near-ring and $I$ be an ideal of $N$. The graph of $N$ with respect to $I$ is a graph with $V (N )$ as vertex set and any two distinct vertices $x$ and $y$ are adjacent if and only if $xNy \subseteq I$ or $yNx \subseteq I$. This graph is denoted by $G_I(N)$. We define the line graph of $G_I(N)$ as a graph with each edge of $G_I (N )$ as vertex and any two distinct vertices are adjacent if and only if their corresponding edges share a common vertex in the graph $G_I (N )$. We denote this graph by $L(G_I (N ))$. We have discussed the diameter, girth, clique number, dominating set of $L(G_I(N))$. We have also found conditions for the graph $L(G_I(N))$ to be acycle graph.


F. Harary, Graph Theory, 1969 by Addison-Wesley Publishing Company, Inc.

H. R. Maimani, M. R. Pournaki and S. Yassemi, Necessary and sufficient conditions for unit graphs to be hamiltonian, Pacific Journal of Mathematics, Vol. 249. No.2, 2011.

K. Chowdhury, Near-rings and near-ring groups with finiteness conditions,VDM Verlag Dr.Muller Aktiengesellschaft and Co.KG, Germany, 2009.

G. Pilz, Near-Rings, North-Holland Publishing Company, Amsterdam. New York. Oxford.1977.

S. Bhavanari, S. P. Kuncham and B. S. Kedukodi, Graph of a near ring with respect to an ideal, Communication in Algebra, 38, (2010), 1957 - 1967.

W. B. Vasantha Kandasamy, Smarandache near-rings, American Reasearch Press, Rehoboth, 2002.




How to Cite

Sarmah, M. ., & Patra, K. (2021). Line Graph Associated to Graph of a Near-Ring with Respect to an Ideal. Tamkang Journal of Mathematics, 52(3), 341-347.