Research Projects

One of the amazing features of twentieth century mathematics is its recognition of the power of the abstract approach. The algebra which has evolved as an outgrowth of all this is not only a subject with an independent life and vigor-it is one of the important current research areas in mathematics, but it is also serves as the unifying thread which interlaces almost all of mathematics such as geometry, number theory, graph theory, analysis, topology and even applied mathematics. Associating a graph with an algebraic object is an active research subject in algebraic graph theory, an area of mathematics in which methods of abstract algebra are employed in studying various graph invariants and tools in graph theory are used in studying various properties of the associated algebraic structure. Research in this subject has attracted considerable attention and has a very long history. For instances, the Cayley graph of a finite group was first considered by Arthur Cayley in 1878 , and Max Dehn in his unpublished lectures on group theory from 1909-1910 reintroduced Cayley graphs of groups under the name Group Diagram, which led to the geometric group theory of today. Research on graphs associated with rings was started in 1988. In 1988, being the first to associate a graph to a ring, Beck introduced and studied the zero-divisor graph of a commutative ring. In 1999, D. F. Anderson and P. S. Livingston modified and studied the zero-divisor graph \Gamma(R) as the graph with vertex set as the non-zero zero-divisors of R and is well-studied in the literature. The idea of k-zero-divisor hypergraph of a commutative ring $R$ was introduced by Ch. Eslachi and A. M. Rahimi in 2007. Actually they extended the concept of zero-divisor of a commutative ring R to that of k-zero-divisor and investigating the interplay between the ring-theoretic properties of R and the hypergraph-theoretic properties of its associated k-uniform hypergraph and later was studied extensively in various research articles. In view of above, there has only one paper published so far about the spectrum of 3-zero divisors in hypergraph from commutative local ring. With this gap, it is proposed to attempt the following problems with respect to the spectrum for the $k$-uniform hypergraphs constructed out of finite commutative rings.  Determine the spectrum of the k-zero-divisor hypergraph of finite commutative ring.  Find the spectrum of the k-maximal ideal hypergraph of finite commutative ring.  Find the various spectrum of k-uniform hypergraph constructed out of finite commutative ring.