Zero-Divisor Graphs of Some Algebraic Structures

This book studies a graph assigned to the zero divisors of a ring with involution * which is an anti-homomorphism of order two. The *-rings with zero-divisor graph connected are characterized and results about chromatic number clique number girth are obtained. An equivalent condition for adjacency in the zero-divisor graph of Rickart *-rings is obtained using the right projections. The zero-divisors graphs of Rickart *-rings are thoroughly investigated using the prime strict spectrum. Also the zero-divisor graphs of dismantlable lattices are examined and are used to obtain the zero-divisor graphs of Rickart *-rings. The zero-divisor graphs of dismantlable lattices are characterized using the comparability graphs and non-ancestor graphs. For two lower dismantlable lattices it is proved that their zero-divisor graphs are isomorphic if and only if the lattices are isomorphic. At last the orthogonality graphs of ortho lattices are investigated and their connection with zero-divisor graphs is established.