Energetic formulation of the subgroup commutativity degree

Finite groups in which every pair of subgroups $(H, K)$ satisfies $HK = KH$ have been classified by Iwasawa, but only in the last decade it was introduced the notion of subgroup commutativity degree $sd(G)$ of groups $G$. From restrictions of numerical nature on $sd(G)$ one usually derives structural conditions on $G$; in fact, among groups $G$ with $sd(G) = 1$, one finds those originally studied by Iwasawa. Here we offer a new perspective of study for $sd(G)$; we use a recently introduced graph, which is called nonpermutability graph of subgroups $Γ_{L(G)}$ of $G$, in order to restrict sd(G) via the notion of energy of $Γ_{L(G)}$ and by means of methods of spectral graph theory. In particular, we find new criteria of nilpotence for $G$ along with new bounds for $sd(G)$.