Zero Divisor Graphs of Some Algebraic Structures: Rickart * Rings and Dismantalabel Lattices

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.