$D^L$, $D^Q$ Spectrum and Eigenvalue-free Interval of Cayley Graphs over Finite Nilpotent Groups

For a finite group $G$ and a symmetric connection set $S$, we study the distance Laplacian ($D^L$) and distance signless Laplacian ($D^Q$) spectrum of the Cayley graph $\text{Cay}(G,S)$ for $G$ a finite nilpotent group. Using the internal direct product decomposition over Sylow subgroups, we reduce every question to a single (possibly non-abelian) prime-power factor. In this setting, the required tools are furnished by ordinary representation theory rather than character theory. Based on this framework, we then derive a complete and explicit formula for every eigenvalue, with multiplicity, of the distance matrix, distance Laplacian, and distance signless Laplacian of $\text{Cay}(G,S)$ for $G$ nilpotent: each eigenvalue is indexed by a tuple of irreducible characters of the Sylow factors. Finally, we use the explicit spectrum, together with Perron–Frobenius theory, to determine the eigenvalue-free interval of $D^L$ and $D^Q$.