فهرست مطالب atossa parsapour

Let $(L,\wedge,\vee)$ be a lattice with a least element $0$. The annihilating-ideal graph of $L$, denoted by $\mathbb{AG}(L)$, is a graph whose vertex-set is the set of all non-trivial ideals of $L$ and, for every two distinct vertices $I$ and $J$, the vertex $I$ is adjacent to $J$ if and only if $I\wedge J=\{0\}$. In this paper, we characterize all lattices $L$ whose the graph $\mathfrak{L}(\mathbb{AG}(L))$ is toroidal.

Given a graph Г, we denote the kth iterated line graph of Г by LK(Г) and LK(Г)=L(LK-1(Г)). In particular, L0(Г)=Г and L1(Г)=L(Г) is the line graph of Г. The toroidality (and projectivity) index of a graph Г is the smallest integer k such that the kth iterated line graph of Г is non-toroidal (and non-projective). We denote the toroidality index of a graph Г by ξT and the projectivity index of a graph Г by ξP. If LK(Г) is toroidal (and projective) for all k≥0, we define ξT=∞ (and ξP=∞). Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a simple graph with vertex set RJ(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. In this paper, we study the toroidality and projectivity indices of Jacobson graphs. We give full characterization of this graph with respect to its toroidality and projectivity indices. Moreover, the toroidality and projectivity index of Jacobson graph is either infinite or two.