Spectral properties of the non--permutability graph of subgroups

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.

کلیدواژه ها:

Subgroup commutativity degree ، Dihedral groups ، Sublattices ، Adjacency Matrix ، Regular graph