@article {
author = {Sharma, A. and Gaur, A.},
title = {Line graphs associated to the maximal graph},
journal = {Journal of Algebra and Related Topics},
volume = {3},
number = {1},
pages = {1-11},
year = {2015},
publisher = {University of Guilan},
issn = {2345-3931},
eissn = {2382-9877},
doi = {},
abstract = {Let $R$ be a commutative ring with identity. Let $G(R)$ denote the maximal graph associated to $R$, i.e., $G(R)$ is a graph with vertices as the elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. Let $\Gamma(R)$ denote the restriction of $G(R)$ to non-unit elements of $R$. In this paper we study the various graphical properties of the line graph associated to $\Gamma(R)$, denoted by $(\Gamma(R))$ such that diameter, completeness, and Eulerian property. A complete characterization of rings is given for which $diam(L(\Gamma(R)))= diam(\Gamma(R))$ orĀ $diam(L(\Gamma(R)))< diam(\Gamma(R))$ or $diam((\Gamma(R)))> diam(\Gamma(R))$. We have shown that the complement of the maximal graph $G(R)$, i.e., the comaximal graph is a Euler graph if and only if $R$ has odd cardinality. We also discuss the Eulerian property of the line graph associated to the comaximal graph.},
keywords = {Maximal graph,line graph,eulerian graph,comaximal graph},
url = {https://jart.guilan.ac.ir/article_1209.html},
eprint = {https://jart.guilan.ac.ir/article_1209_6febd2a7a22b03870dcd02ddde00b032.pdf}
}