A study on rough ideal based rough edge cayley graph

Abstract

In this thesis, we have considered an approximation space I = (U,R) where U is a non empty finite set of objects called the universe and R is an equivalence relation defined on U. Let (T,and#8710;,and#8711;) be the Rough semiring induced by I = (U,R), where T = {RS(X) | X and#8838; U}. Let {X1,X2,...,Xn} be the nand#8722;equivalence classes induced by R. Without loss of generality we assume that {X1,X2,...,Xm} be the mand#8722;equivalence classes whose cardinality is greater than one and {Xm+1,Xm+2,...,Xn} be the nand#8722;m equivalence classes whose cardinality is equal to one. Let B = {xi | xi and#8712; Xi , |Xi | gt 1} be the set of pivot elements chosen from the equivalence classes whose cardinality is greater than one and J = {RS(X) | X and#8712; P(B)} is an ideal called the Rough ideal of the Rough semiring (T,and#8710;,and#8711;). Several work has been done in studying this Rough semiring (T,and#8710;,and#8711;). In this work, the concept of Rough Ideal J is used in defining a Cayley graph called the Rough Ideal based Rough Edge Cayley Graph, G(T(J)) whose vertices are elements of T and |T| = 2 nand#8722;m(3 m) , where 0 lt m and#8804; n, n and m are positive integers. Any two vertices RS(X) and RS(Y) are connected in G(T(J)) if RS(X)and#8711;RS(Z) = RS(Y), RS(Z) and#8712; J. In G(T(J)), all the vertices will be connected to RS(and#966;). The graph obtained by removing these trivial edges is G(T(Jb)). The complexity in studying G(T(J)) will increase for large values of n and m. In this study, the complexity is made simpler by defining a category graph CG corresponding to G(T(J)) in which the vertices are partitioned into disjoint categories in such a way that all elements belonging to a particular category will behave similarly. By this we made a detailed study in finding degree of each vertex of a particular category and cardinality of each category in G(T(J)) and G(T(Jb)). Also the wiener index, domination number, maximal independent set, independent domination number, chromatic number, chromatic index, diameter, rainbow connection number, clique number and girth of G(T(J)) and G(T(Jb)) are also studied in detail. All the developed conce