Geodesic
Zero-divisor graphs of reduced Rickart *-rings
Discussiones Mathematicae. General Algebra and Applications, Tome 37 (2017) no. 1, pp. 31-43

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For a ring A with an involution *, the zero-divisor graph of A, Γ*(A), is the graph whose vertices are the nonzero left zero-divisors in A such that distinct vertices x and y are adjacent if and only if xy* = 0. In this paper, we study the zero-divisor graph of a Rickart *-ring having no nonzero nilpotent element. The distance, diameter, and cycles of Γ*(A) are characterized in terms of the collection of prime strict ideals of A. In fact, we prove that the clique number of Γ*(A) coincides with the cellularity of the hullkernel topological space Σ(A) of the set of prime strict ideals of A, where cellularity of the topological space is the smallest cardinal number m such that every family of pairwise disjoint non-empty open subsets of the space have cardinality at most m.
Keywords: reduced ring, Rickart *-ring, zero-divisor graph, prime strict ideals 
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Patil, A.A.; Waphare, B.N. Zero-divisor graphs of reduced Rickart *-rings. Discussiones Mathematicae. General Algebra and Applications, Tome 37 (2017) no. 1, pp. 31-43. http://geodesic.mathdoc.fr/item/DMGAA_2017_37_1_a2/