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

@misc{manna2026spectrum,
  title = {{$D^L$, $D^Q$ Spectrum and Eigenvalue-free Interval of Cayley Graphs over Finite Nilpotent Groups}},
  author = {Pallabi Manna and Santanu Mandal},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13429},
  eprint = {2610.13429},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13429},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - D^L, D^Q Spectrum and Eigenvalue-free Interval of Cayley Graphs over Finite Nilpotent Groups
AU  - Pallabi Manna
AU  - Santanu Mandal
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13429
UR  - https://arxiv.org/abs/2610.13429
AB  - 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 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 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.
ER  -