MULTIPLE CO-PRIME DIVISORS AND MULTIPLE DIVISORS GRAPHS

Abstract

In this paper we introduce two new concepts of graphs. Let R be a commutative ring with identity and Z(R)\{0} be the set of elements in R  divided  it into two sets Z₁(R) and Z₂(R) , where Z₁(R) be the set of co-prime divisor elements satisfies the Euler’s function and Z₂(R) be the set of non-co-prime divisor elements, a simple graph MC(R) is associated to Z(R) it is called multiple co-prime divisors graph. Moreover, Let R be a commutative ring with identity and Z*(R)=Z(R)\{0} be the set of all non-zero divisor elements in R, a simple graph M(R) is associated to R for distinct elements a and b in Z*(R) is an edge in M(R) if and only if aba=a(mod n) for all n in N. Also, diameter, girth, chromatic number and nullity of multiple co-prime divisor and multiple divisor graphs will be determined.