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 G be a group. Recall that the intersection graph of subgroups of G is an undirected graph whose vertex set is the set of all nontrivial subgroups of G and distinct vertices H, K are joined by an edge in this graph if and only if the intersection of H and K is nontrivial. The aim of this article is to investigate the interplay between the group-theoretic properties of a finite group G and the graph-theoretic properties of the Complement of the intersection graph of subgroups of G.

Let 𝑅 be a commutative ring with identity and 𝑀 be a unitary 𝑅-module, and let 𝐼 (𝑅 )* be the set of all nontrivial ideals of 𝑅 . The Complement of the 𝑀-intersection graph of ideals of 𝑅 , denoted by Γ (𝑅 ), is a graph with the vertex set 𝐼 (𝑅 )*, and two distinct vertices 𝐼 and 𝐽 are adjacent if and only if 𝐼 𝑀 ∩ 𝐽 𝑀 ={0}. In this paper, for every multiplication 𝑅-module 𝑀 , the diameter and the girth of Γ (𝑅 ) are determined. Also, we show that if 𝑚 , 𝑛 >1 are two integers and ℤ 𝑛 is a ℤ 𝑚-module, then the Complement of the ℤ 𝑛-intersection graph of ideals of ℤ 𝑚 is weakly perfect.

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 $CI(R)$.

Let G be a finite group with the identity e. The subgroup intersection graph  GSI (G) of G is the graph with vertex set V (GSI (G)) =G-e and two distinct vertices x and y are adjacent in GSI (G) if and only if |áxñՈáyñ|>1, where áxñ is the cyclic subgroup of G generated by xÎG. In this paper, we obtain a lower bound for the independence number of subgroup intersection graph. We characterize certain classes of subgroup intersection graphs corresponding to finite abelian groups. Finally, we characterize groups whose automorphism group is the same as that of its subgroup intersection graph.

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 $G(A)$.