Abstract
Let R be a commutative ring with 1. In Biswas et al. (Disc Math Algorithms Appl 11(1):1950013, 2019), we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if \(aR+bR=eR\) for some nonzero idempotent e in R. Let \(G'(R)\) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph \(G'(R)\) is connected and Eulerian. Also we characterize those rings R for which genus of the graph \(G'(R)\) is \(\le 2\). Finally, we show that the graph \(G'(R)\) is a line graph of some graph if and only if R is either a regular ring or a local ring.
Highlights
The algebraic graph theory is an interesting subject for graph theorists as well as algebraists, since it relates two different areas of mathematics, involves some combinatorial approach and can be studied different levels of mathematical expertise and sophistication
The interplay between ring theoretic and graph theoretic properties was studied by several authors and this approach has since become increasingly very popular in abstract algebraic graph theory
On the other hand after translating algebraic properties of rings into graph theoretic language, difficult problems in ring theory might be more solved by using techniques from graph theory
Summary
The algebraic graph theory is an interesting subject for graph theorists as well as algebraists, since it relates two different areas of mathematics, involves some combinatorial approach and can be studied different levels of mathematical expertise and sophistication. Rk be a finite commutative ring with identity such that each Ri is a local ring with unique maximal ideal Mi. Let a = (a1 , a2 , ..., ak ) be an element of R1 deg(a) in G(R) is either. Let R be a finite commutative ring with 1 It follows from [7], there exist local rings R1 , R2 , ..., Rk with unique maximal ideals M1 , M2 , ..., Mk respectively such that R = R1 ⇥ R2 ⇥ ... Let R be a finite local ring and M be its unique maximal ideal. Let R(6= F8 ) be a local ring of order 8 and M be its unique maximal ideal.
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