ULethbridge - Number Theory and Combinatorics Seminar: Angsuman Das

Associating graphs with groups date back to Arthur Cayley. In this lecture, we will discuss about another graph, called co-maximal subgroup graph Γ(G) introduced by Akbari et al. in [1]. The co-maximal subgroup graph Γ(G) of a group G whose vertices are non-trivial proper subgroups of G and two vertices H and K are adjacent if HK=G. Though the definition allows the possibility of G to be infinite, in our discussion, we will focus mainly on finite groups. We discuss various graph parameters like diameter, connectedness, existence of isolated vertices, girth, bipartiteness etc. Finally, we will highlight some problems on realizability and graph isomorphisms, and some partial solutions to those questions in terms of properties of G. Most of the results presented in the lecture can be found in [1] and [2].

References:

1. S. Akbari, B. Miraftab and R. Nikandish: Co-maximal Graphs of Subgroups of Groups, Canadian Math Bulletin, Vol. 60(1), pp.12-25, 2017.
2. A. Das, M. Saha and S. Alkaseasbeh: On Co-Maximal Subgroup Graph of a Group, https://arxiv.org/abs/2103.14284