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

**Author**

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

**Abstract**

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.

**Keywords**

**Main Subjects**

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Introduction to the Proceedings of WGTTG2021

September 2022Pages 281-294

**Receive Date:**12 August 2021**Revise Date:**02 March 2022**Accept Date:**01 April 2022**Published Online:**01 September 2022