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Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
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Why do we associate a graph to a ring?
I don't know about the papers you link to specifically, but here is one way to justify introducing graphs into any object you want.
In particular, graphs are useful structures for encoding interaction …