A study on generalized total graphs from commutative rings

Abstract

There are lot of ways to associate graphs with algebraic structures. newlineSome of them to mention are Cayley graphs from groups, zero-divisor newlinegraphs and total graphs from commutative rings. The idea of associating newlinea graph with zero-divisors of a commutative ring was introduced by newlineBeck in 1988. Also Beck has investigated the interplay between the ringtheoretic newlineproperties of a commutative ring and the associated zero-divisor newlinegraph. Further Anderson and Badawi introduced the concept total graph newlineof commutative rings in the year 2008. The total graph of a commutative newlinering R is the undirected graph with R as the vertex set and two distinct newlinevertices in R are adjacent if and only if their sum is a zero-divisor of newlineR. Recently Anderson and Badawi introduced and studied the generalized newlinetotal graph of commutative rings with respect to the multiplicatively newlineprime subset H of R. The generalized total graph of a commutative ring newlineis the undirected graph with all elements of R as vertices, and for two newlinedistinct vertices in R are adjacent if and only if their sum is in H. In this newlinedissertation, an attempt has been made to study about graph theoretical newlineproperties and various domination parameters of generalized total graph newlineof commutative rings and its complement. newline