Let R be a commutative ring and M be an R-module. The M-Intersection graph of ideals of R, denoted by GM(R) is a graph with the vertex set I(R) ∗, , and two distinct vertices I and J are adjacent if and only if IM ∩,JM ̸, = 0. In this paper, we study GR/J (R/I), where I and J are ideals of R and I ⊆,J. We characterize all ideals I and J for which GR/J (R/I) is planar, outerplanar or ring graph.

The comaximal Intersection graph $CI(R)$ of ideals of a ring $R$ is an undirected graph whose vertex set is the collection of all non-trivial (left) ideals of $R$ and any two vertices $I$ and  $J$ are adjacent if and only if $I+J=R$ and $I\cap J\neq0$. We study the connectedness of $CI(R)$. We also discuss  independence number, clique number, domination number, chromatic number of $CI(R)$.

For any non-empty set and k-subset  , the k-Intersection graph denoted by 􀀀 m(;  ) is undirected simple graph whose vertices are all m-subsets of and two distinct vertices A and B are adjacent if and only if A \ B *  . In this paper, we determine diameter, girth, some numerical invariants and planarity, hamiltonian and perfect matching of these graphs. Moreover adjacency matrix is considered at the end.

For a ring R, the Intersection minimal ideal graph, denoted by ∧(R), is a simple undirected graph whose vertices are proper non-zero (right) ideals of R and any two distinct vertices I1 and I2 are adjacent if and only if I1 ∩ I2 is a minimal ideal of R. In this article, we explore connectedness, clique number, split character, planarity, independence number, domination number of ∧(R).

In this paper, we define the co-Intersection graph $G(A)$ of an \(S\)-act \(A\) which is a graph whose vertices are non-trivial subacts of \(A\) and two  distinct  vertices \(B_1\) and \( B_2\) are adjacent if \(B_1 \cup B_2\neq A\).  We investigate the relationship between the algebraic properties of an  \(S\)-act \(A\) and the properties of the graph $G(A)$.

Let $R$ be a commutative ring with unity and $A(R)$ be the set of annihilating-ideals of $R$. The annihilator Intersection graph of $R$, represented by $AIG(R)$, is an undirected graph with $A(R)^*$ as the vertex set and $\mathfrak{M} \sim \mathfrak{N}$ is an edge of $AIG(R)$ if and only if $Ann(\mathfrak{M}\mathfrak{N}) \neq Ann(\mathfrak{M}) \cap Ann(\mathfrak{N})$, for distinct vertices $\mathfrak{M}$ and $\mathfrak{N}$ of $AIG(R)$. In this paper, we first defined finite commutative rings whose annihilator Intersection graph is isomorphic to various well-known graphs, and then all finite commutative rings with a planar or toroidal annihilator Intersection graph were characterized.

Let M be a unitary left R-module, where R is a (not necessarily commutative) ring with identity. The small Intersection graph of nontrivial submodules of M, denoted by 􀀀, (M), is an undirected simple graph whose vertices are in one-to-one correspondence with all nontrivial submodules ofM and two distinct vertices are adjacent if and only if the Intersection of corresponding submodules is a small submodule of M. In this paper, we investigate the fundamental properties of these graphs to relate the combinatorial properties of ℾ, (M) to the algebraic properties of the module M. We determine the diameter and the girth of ℾ, (M). We obtain some results for connectivity and planarity of these graphs. Moreover, we study orthogonal vertex, domination number and the conditions under which the graph ℾ, (M) is complemented.