2015/05/29 by R. Rajkumar, Rajkumar, R., P. Devi +1
Mathematics · #05C10 #05C25 #05E15 #20E99 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:05C10 #msc:05C25 #msc:05E15 #msc:20E99
paper · pdf · doi:10.48550/arxiv.1505.08094
38 pages, 10 figures. arXiv admin note: text overlap with arXiv:1505.03462, some corrections made
arxiv created 2015/06/02 · arxiv updated 2015/06/03
Let G be a group. The intersection graph of subgroups of G, denoted by \mathscrI(G), is a graph with all the proper subgroups of G as its vertices and two distinct vertices in \mathscrI(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 K5, K4, C5, C4, P4, P3, P2, K1,3, K2,3 or K1,4 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.