Complete Laplacian Spectra and Rigidity of Enhanced Power Graphs

Abstract

Let p be prime, a>=2, m>=2, and gcd(p,m)=1. Using the Chinese remainder theorem, we study the enhanced power graph of G_{a,p,m}=(C_{p^a} x C_p) x C_m. The mixed-order family is the lexicographic product Gamma_{a,p}[K_m], where Gamma_{a,p}=E(C_{p^a} x C_p). This gives a two-stage compression mp^{a+1}->p^{a+1}->2a and yields the complete Laplacian spectrum together with signless- and normalized-Laplacian transfers. For the maximum- and minimum-degree weighted adjacency matrices M_max and M_min, we identify a common spectral submultiset of size N-2a, N=mp^{a+1}. We further prove that this overlap is exact for a=2 and a=3: the corresponding residual matrices have disjoint spectra, so the common multiplicities are N-4 and N-6, respectively. The a=3 proof uses a finite exact-arithmetic interpolation certificate, stated and justified in Appendix A so that the argument is contained entirely in the article. Finally, the Laplacian spectrum determines (p,a,m) within the family and imposes rigidity on Laplacian-cospectral mates.