The complement of the intersection graph of subgroups of a group G, denoted by 5c(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 & AND; K = 1, where 1 denotes the trivial subgroup of G. 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 C3-free. Also we characterize all the finite groups whose complement of the intersection graph of subgroups is planar.

In this paper, we study the structure of the permutability graphs of subgroups, and the permutability graphs of non-normal subgroups of the following groups: the dihedral groups D-n, the generalized quaternion groups Qn, the quasi-dihedral groups QD(2n) and the modular groups M-pn. Further, we investigate the number of edges, degrees of the vertices, independence number, dominating number, clique number, chromatic number, weakly perfectness, Eulerianness, Hamiltonicity of these graphs.

Let G be a group. The permutability graph of subgroups of G, denoted by (G), is a graph having all the proper subgroups of G as its vertices, and two subgroups are adjacent in (G) if and only if they permute. In this article, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of K-1,K- 5, P-5, P-6, C-6, or K-3,K- 3 as a subgraph.

Let G be a group. The intersection graph of cyclic subgroups of G, denoted by Ic(G), is a graph having all the proper cyclic subgroups of G as its vertices and two distinct vertices in Ic(G) are adjacent if and only if their intersection is non-trivial. In this paper, we classify the finite groups whose intersection graphs of cyclic subgroups are one of totally disconnected, complete, star, path, cycle. We show that for a given finite group G, girth(Ic(G))∈{3,∞}. Moreover, we classify all finite non-cyclic abelian groups whose intersection graphs of cyclic subgroups are planar. Also for any group G, we determine the independence number, clique cover number of Ic(G) and show that Ic(G) is weakly α-perfect. Among the other results, we determine the values of n for which Ic(Zn) is regular and estimate its domination number.

For a finite group $G$, we define the inclusion graph of subgroups of $G$, denoted by $\mathcal I(G)$, is a graph having all the proper subgroups of $G$ as its vertices and two distinct vertices $H$ and $K$ in $\mathcal I(G)$ are adjacent if and only if either $H \subset K$ or $K \subset H$. In this paper, we classify the finite groups whose inclusion graph of subgroups is one of complete, bipartite, tree, star, path, cycle, disconnected, claw-free. Also we classify the finite abelian groups whose inclusion graph of subgroups is planar. For any given finite group, we estimate the clique number, chromatic number, girth of its inclusion graph of subgroups and for a finite abelian group, we estimate the diameter of its inclusion graph of subgroups. Among the other results, we show that some groups can be determined by their inclusion graph of subgroups

The intersection graph of subgroups of a group G is a graph whose vertex set is the set of all proper subgroups of G and two distinct vertices are adjacent if and only if their intersection is non-trivial. In this paper, we obtain the clique number and degree of vertices of intersection graph of subgroups of dihedral group, quaternion group and quasi-dihedral group.

Let G be a group. The permutability graph of cyclic subgroups of G, denoted by Γ_c(G), is a graph with all the proper cyclic subgroups of G as its vertices and two distinct vertices in Γ_c(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graph of cyclic subgroups belongs to one of the following: bipartite, tree, star graph, triangle-free, complete bipartite, P_n, C_n, K_4, K_1,3-free, unicyclic. We classify abelian groups whose permutability graph of cyclic subgroups are planar. Also we investigate the connectedness, diameter, girth, totally disconnectedness, completeness and regularity of these graphs.

Let G be a group. The intersection graph of subgroups of G, denoted by ℐ(G), is a graph with all the proper subgroups of G as its vertices and two distinct vertices in ℐ(G) are adjacent if and only if the corresponding subgroups having a non-trivial intersection in G. In this paper, we classify the finite groups whose intersection graph of subgroups are toroidal or projective-planar. In addition, we classify the finite groups whose intersection graph of subgroups are one of bipartite, complete bipartite, tree, star graph, unicyclic, acyclic, cycle, path or totally disconnected. Also we classify the finite groups whose intersection graph of subgroups does not contain one of K_5, K_4, C_5, C_4, P_4, P_3, P_2, K_1,3, K_2,3 or K_1,4 as a subgraph. We estimate the girth of the intersection graph of subgroups of finite groups. Moreover, we characterize some finite groups by using their intersection graphs. Finally, we obtain the clique cover number of the intersection graph of subgroups of groups and show that intersection graph of subgroups of groups are weakly α-perfect.

Let $G$ be a group. We define the coprime graph of subgroups of $G$, denoted by $\mathcal P(G)$, is a graph whose vertex set is the set of all proper subgroups of $G$, and two distinct vertices are adjacent if and only if the order of the corresponding subgroups are coprime. In this paper, we study some connections between algebraic properties of a group and graph theoretic properties of its coprime graph.

The permutability graph of subgroups of a given group G, denoted by Γ(G), is a graph with vertex set consists of all the proper subgroups of G and two distinct vertices in Γ(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graphs of subgroups are one of bipartite, star graph, C3-free, C5-free, K4-free, K5-free, K1,4-free, K2,3-free or Pn-free (n = 2, 3, 4). We investigate the same for infinite groups also. Moreover, some results on the girth, completeness and regularity of the permutability graphs of subgroups of groups are obtained. Among the other results, we characterize groups Q8, S3 and A4 by using their permutability graphs of subgroups.