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The zero-divisor graph of a ring with involution

Department of Mathematics, Garware College of Commerce, Pune-411004, India

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Center for Advanced Studies in Mathematics, Department of Mathematics, Savitribai Phule Pune University, Pune-411007, India

E-mail Address: bnwaph@math.unipune.ac.in

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https://doi.org/10.1142/S0219498818500500Cited by:6 (Source: Crossref)

Abstract

For a *-ring $A$ , we associate a simple undirected graph $Γ^{*} (A)$ having all nonzero left zero-divisors of $A$ as vertices and, two vertices $x$ and $y$ are adjacent if $x y^{*} = 0$ . In case of Artinian *-rings and Rickart *-rings, characterizations are obtained for those *-rings having $Γ^{*} (A)$ a complete graph or a star graph, and sufficient conditions are obtained for $Γ^{*} (A)$ to be connected and also for $Γ^{*} (A)$ to be disconnected. For a Rickart *-ring $A$ , we characterize the girth of $gr (Γ^{*} (A))$ and prove a sort of Beck’s conjecture.

Communicated by P. Ara

Keywords:

AMSC: 16W10, 05C25, 05C15

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