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Given a finite group G and the subgroups lattice L(G) of G, the \textit{non--permutability graph of subgroups} ΓL(G) is introduced as the graph with vertices in L(G)\CL(G)(L(G)), where CL(G)(L(G)) is the smallest sublattice of L(G) containing all permutable subgroups of G, and edges obtained by joining two vertices X,Y if XY≠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. [ABSTRACT FROM AUTHOR]