摘要
For a finite group G,the co-maximal subgroup graphΓ(G)of G is a graph whose vertices are proper subgroups of G,and two distinct vertices H and K are adjacent if and only if H K=G.The deleted co-maximal subgroup graphΓ^(∗)(G)is obtained by removing isolated vertices fromΓ(G).Firstly,we provide necessary and sufficient conditions forΓ^(∗)(G)to be connected;in particular,from the viewpoint of normal subgroups in G,we give some sufficient conditions forΓ^(∗)(G)to be connected.Secondly,for a finite abelian group G we prove that the diameter ofΓ^(∗)(G),diam(Γ^(∗)(G)),is at most 3.Also,we characterize G with diam(Γ^(∗)(G))=i for i=1,2,3 and we give characterizations for G withΓ^(∗)(G)being complete bipartite graphs and null graphs separately.Finally,we show that for the semidirect product G of two finite cyclic groups,Γ^(∗)(G)is connected and diam(Γ^(∗)(G))=2.
基金
supported by National Natural Science Foundation of China(Grant Nos.12071194,12361072)
2023 Xinjiang Uygur Autonomous Region Natural Science Foundation General Project(No.2023D01A36)
2023 Xinjiang Uygur Autonomous Region Natural Science Foundation for Youths(No.2023D01B48).
