The Complement of the intersection graph of subgroups of a group $G$, denoted by $\mathcal{I}^c(G)$, is the graph whose vertex set is the set of all nontrivial proper subgroups of $G$ and its two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K$ is trivial. In this paper, we classify all finite groups whose Complement of the intersection graph of subgroups is one of totally disconnected, bipartite, complete bipartite, tree, star graph or $C_3$-free. Also we characterize all the finite groups whose Complement of the intersection graph of subgroups is planar.

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 $G(A)$.