@article{Raza2026, 
author = {Muhammad Awais Raza and Muhammad Khalid Mahmood and Daniele Ettore Otera},
title = {Structural properties of generalized power congruence graphs over sets of moduli},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {17564-17583},
keywords = {power congruence graphs, prime power moduli, graph decomposition, spectral graph theory, Laplacian matrix, spectra},
url = {https://www.sciopen.com/article/10.3934/math.2026718},
doi = {10.3934/math.2026718},
abstract = {In this article, we introduce and study a novel class of graphs called power congruence graphs (PCGs) that are constructed over the sets of moduli of the form        M    p    =  {      p    t    :  t  ≥  1  ,        p    t    <  n  }, where    p is a prime. For    n  ∈            Z              +      , consider    V  =  {  0  ,  1  ,  …  ,  n  −  1  } as the vertex set. We construct a simple, undirected graph    G  (  n  ,  k  ,      M    p    ) without loops or multiple edges over    V in which two distinct vertices    a  ,  b  ∈  V are adjacent if        a    k    ≡  b    (  mod      m    ) for some    m  ∈      M    p   and fixed    k  ∈            Z        +  . We present a comprehensive structural characterization of PGCs for the cases    p  =  2  ,  3  ,  5 and extend the framework to an arbitrary prime    p. When    p  =  2, the graph decomposes into two disjoint complete components for all    k. When    p  =  3, the graph structure is governed by    k  mod  2; for odd value of    k, the graph is a disjoint union of three complete components; and for even value of    k, the graph is a disjoint union of one complete component and one component        K    n          F       obtained from a complete graph        K    n   by deleting a specified set of edges    F  ⊆  E  (      K    n    ). When    p  =  5, the graph becomes more intricate and depends on    k  mod  4, producing configurations that include both complete components and components        K    n          F       obtained from a complete graph        K    n   by deleting a specified set of edges    F  ⊆  E  (      K    n    ). In general, for a prime    p, the structure of the graph is determined by the residue class of    k  mod  (  p  −  1  ), giving rise to up to    p  −  1 distinct structural types. This highlights a systematic transition from simple to increasingly complex graph configurations as the prime modulus increases. Furthermore, we investigate several graph invariants associated with these graphs. This study provides a framework for understanding power congruence-based graph constructions bridging number theory with graph theory.}
}