Factorization number and subgroup commutativity degree via spectral invariants

The factorization number $F_2(G)$ of a finite group $G$ is the number of all possible factorizations of $G=HK$ as product of its subgroups $H$ and $K$, while the subgroup commutativity degree $\mathrm{s}\mathrm{d}(G)$ of $G$ is the probability of finding two commuting subgroups in $G$ at random. It is known that $\mathrm{s}\mathrm{d}(G)$ can be expressed in terms of $F_2(G)$. Denoting by $\mathrm{L}(G)$ the subgroups lattice of $G$, the non–permutability graph of subgroups ${\mathrm{\Gamma }}_{\mathrm{L}(G)}$ of $G$ is 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 $X,Y$ such that $XY\mathrm{\ne }YX$. The spectral properties of ${\mathrm{\Gamma }}_{\mathrm{L}(G)}$ have been recently investigated in connection with $F_2(G)$ and $\mathrm{s}\mathrm{d}(G)$. Here we show a new combinatorial formula, which allows us to express $F_2(G)$, and so $\mathrm{s}\mathrm{d}(G)$, in terms of adjacency and Laplacian matrices of ${\mathrm{\Gamma }}_{\mathrm{L}(G)}$.