THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-II

Abstract

The intersection graph I TΓ (R) of gamma sets in the total graph T Γ (R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about I TΓ (R) where R is a commutative Artin ring. In this paper, we continue our interest on I TΓ (R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of I TΓ (R). Further, we focus on certain graph theoretic parameters of I TΓ (R) like the independence number, the clique number and the connectivity of I TΓ (R). Also, we obtain both vertex and edge chromatic numbers of I TΓ (R). In fact, it is proved that if R is a finite commutative ring, then χ(I TΓ (R)) = ω(I TΓ (R)). Having proved that I TΓ (R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which I TΓ (R) is perfect. In this sequel, we characterize all commutative Artin rings for which I TΓ (R) is of class one (i.e. χ′(I TΓ (R)) = Δ(I TΓ (R))). Finally, it is proved that the vertex connectivity and edge connectivity of I TΓ (R) are equal to the degree of any vertex in I TΓ (R).