Spectral properties of the non--permutability graph of subgroups

Given a finite group G and the subgroups lattice L(G) of G, the extit {non--permutability graph of subgroups} is introduced as the graph with vertices in L(G), where is the smallest sublattice of LG containing all permutable subgroups of , and edges obtained by joining two vertices X,Y if XY≭YX if . 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.