Graph-Theoretic Invariants and the Normal Subgroup Structure of Automorphism Groups A Unified Treatment of Spectral, Chromatic, and Topological Perspectives Mathematics Subject Classification (2020): 05C25, 05C50, 05E18, 20B25, 20E15 (مجلة الفاروق للعلوم, 2026)
This paper offers a unified, research-level treatment of the interaction between graph-theoretic invariants and the normal subgroup structure of graph automorphism groups. Starting from the classical framework of Frucht, Sabidussi, Whitney and Watkins, we develop the analysis of  as a permutation group through three complementary lenses: (i) the spectral lens, where irreducible representations of  decompose the eigenspaces of the adjacency and Laplacian operators; (ii) the chromatic–combinatorial lens, connecting the chromatic number, distinguishing number, and girth to orbit and stabilizer structure; and (iii) the topological lens, in which the genus of  constrains the possible normal subgroup lattice of  via covering-space and voltage-graph techniques.The principal contribution of this paper is the development of a unified theoretical framework linking graph-theoretic invariants with the normal subgroup structure of graph automorphism groups. Within this framework, three principal synthesis theorems are established, integrating spectral, chromatic, and topological perspectives into a common structural interpretation supported by representative examples and detailed proofs.