%0 Journal Article
%T Line graphs associated to the maximal graph
%J Journal of Algebra and Related Topics
%I University of Guilan
%Z 2345-3931
%A Sharma, A.
%A Gaur, A.
%D 2015
%\ 06/01/2015
%V 3
%N 1
%P 1-11
%! Line graphs associated to the maximal graph
%K Maximal graph
%K line graph
%K eulerian graph
%K comaximal graph
%R
%X 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.
%U https://jart.guilan.ac.ir/article_1209_6febd2a7a22b03870dcd02ddde00b032.pdf