Name: Shashikumar R
Country: India

Year: 2025
Volume: Volume-10, Issue-2 (July-December)
Page Number: 475-486
DOI: https://doi.org/10.5281/zenodo.22830312

The interaction between group theory and graph theory has developed from classical representations of algebraic structures into a broad interdisciplinary research domain encompassing algebraic graph theory, spectral analysis, computational group theory, network science, quantum information and symmetry-aware artificial intelligence. This article presents a systematic and bibliometric review of research connecting group-theoretic structures with graphs, with emphasis on Cayley, Schreier, power, commuting and non-commuting graphs, graph automorphisms, spectral methods and representation-theoretic approaches. It further examines the transition of these mathematical structures toward computational and interdisciplinary applications, including complex networks, biomedical systems, cryptography, quantum computing, geometric deep learning, molecular modelling and materials science. The review combines systematic-review principles, mathematical taxonomy and a reproducible bibliometric protocol. The evidence window is restricted to literature published or publicly available on or before 31 May 2025. The synthesis shows that algebraic graph theory operates as a bidirectional interface: algebraic structures generate graphs, graph properties reveal algebraic structure, and matrix- and representation-theoretic methods mediate between the two. The paper proposes an evolutionary framework progressing from group structure and group actions through graph symmetry, algebraic graph construction, computational analysis, network representation and symmetry-aware learning. It identifies research opportunities in approximate symmetry, dynamic algebraic graphs, scalable computation, mathematically interpretable graph learning, quantum graph computation and scientific AI. A six-layer Algebraic Graph Intelligence framework is proposed to connect algebraic foundations with intelligent graph-based applications.

Keywords: group theory; algebraic graph theory; Cayley graphs; Schreier graphs; power graphs; commuting graphs; graph automorphism; spectral graph theory; representation theory; bibliometrics; graph neural networks; equivariant learning; quantum computing; network science