2024/11/07 by Malviy, Siddharth, Kakkar, Vipul
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.04495
The commuting graph Γ(G) of a group G is the simple undirected graph with group elements as a vertex set and two elements x and y are adjacent if and only if xy=yx in G. By eliminating the identity element of G and all the dominant vertices of Γ(G), the resulting subgraphs of Γ(G) are Γ^*(G) and Γ**(G), respectively. In this paper, we classify all the finite groups G such that the graph Δ(G) ∈ \Γ(G), Γ^*(G), Γ**(G)\ is the line graph of some graph. We also classify all the finite groups G whose graph Δ(G) ∈ \Γ(G), Γ^*(G), Γ**(G)\ is the complement of line graph.