Power Congruence Graphs over Two Coprime Moduli: Symmetry of the Rook Graph Blow-Up and Spectral Invariants

We study the undirected power congruence graph Gnk({p,q}) on vertex set V={0,1,…,n−1} in which two distinct vertices a,b are adjacent whenever ak≡bk(modp) or ak≡bk(modq), where p,q are distinct primes with pq<n and k≥1 is fixed. While the single-prime moduli set {p,p2,…,pt} decomposes into a disjoint union of complete graphs, we show that two coprime prime moduli produce a fundamentally different structure: the non-degenerate part of the graph is a blow-up of the rook graph on a dp×dq grid, where dp=(p−1)/gcd(k,p−1) and dq=(q−1)/gcd(k,q−1). We give explicit block sizes via the Chinese Remainder Theorem, describe the three degenerate residue classes V00,Vp0,V0q and their gluing to the main rook graph, and derive the adjacency and Laplacian spectra through an equitable partition. Explicit formulas for the Kirchhoff index, Estrada index and graph energy in terms of the spectrum of a quotient matrix of size (dp+1)(dq+1) follow. Three worked examples (n=30,p=2,q=3,k=2; n=20,p=3,q=5,k=2; n=100,p=3,q=7,k=3) illustrate the uniform and non-uniform cases. The row/column permutation symmetry of the base grid is inherited by the blow-up and underlies the equitable partition.