Line graphs associated to the maximal graph

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)) or 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