Summary: The intersection graph \(I_{T\Gamma}(R)\) of gamma sets in the total graph \(I_{\Gamma}(R)\) of a commutative ring \(R\), is the undirected graph with vertex set as the collection of all \(\gamma\)-sets in the total graph of \(R\) and two distinct vertices \(u\) and \(v\) are adjacent if and only if \(u\cap v\neq \emptyset\).
T. T. Chelvam and T. Asir [J. Algebra Appl. 12, No. 4, Paper No. 1250198, 18 p. (2013; Zbl 1264.05061)] investigated \(I_{T\Gamma}(R)\) where \(R\) is a commutative Artin ring.
In this paper, we continue to investigate \(I_{T\Gamma}(R)\) and actually we study the Eulerian, Hamiltonian and pancyclic nature of \(I_{T\Gamma}(R)\). Further, we focus on certain graph theoretic parameters of \(I_{T\Gamma}(R)\) like the independence number, the clique number and the connectivity of \(I_{T\Gamma}(R)\). Also, we obtain both vertex and edge chromatic numbers of \(I_{T\Gamma}(R)\).
In fact, it is proved that if \(R\) is a finite commutative ring, then \(\chi (I_{T\Gamma}(R)) = \omega(I_{T\Gamma}(R))\). Having proved that \(I_{T\Gamma}(R)\) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which \(I_{T\Gamma}(R)\) is perfect.
In this sequel, we characterize all commutative Artin rings for which \(I_{T\Gamma}(R)\) is of class one (i.e., \(\chi^{\prime}(I_{T\Gamma}(R)) = \Delta(I_{T\Gamma}(R)))\).
Finally, it is proved that the vertex connectivity and edge connectivity of \(I_{T\Gamma}(R)\) are equal to the degree of any vertex in \(I_{T\Gamma}(R)\).