Spectral properties of the non--permutability graph of subgroups

Document Type : Workshop on Graphs, Topology and Topological Groups, Cape Town, South Africa


Department of Mathematics and Applied Mathematics, Faculty of Science, University of Cape Town, South Africa.


Given a finite group $G$ and the subgroups lattice $\mathrm{L}(G)$ of $G$, the \textit{non--permutability graph of subgroups} $\Gamma_{\mathrm{L}(G)}$ is introduced as the graph with vertices in $\mathrm{L}(G) \setminus \mathfrak{C}_{\mathrm{L}(G)}(\mathrm{L}(G))$, where $\mathfrak{C}_{\mathrm{L}(G)}(\mathrm{L}(G))$ is the smallest sublattice of $\mathrm{L}(G)$ containing all permutable subgroups of $G$, and edges obtained by joining two vertices $X,Y$ if $XY\neq YX$. Here we study the behaviour of the non-permutability graph of subgroups using algebraic properties of associated matrices such as the adjacency and the Laplacian matrix. Further, we study the structure of some classes of groups whose non-permutability graph is strongly regular.


Main Subjects

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Volume 11, Issue 3 - Serial Number 3
Introduction to the Proceedings of WGTTG2021
September 2022
Pages 281-294
  • Receive Date: 12 August 2021
  • Revise Date: 02 March 2022
  • Accept Date: 01 April 2022
  • Published Online: 01 September 2022