Graph theory provides a powerful mathematical framework for representing and analyzing the structural and topological properties of digital images. A digital image can naturally be modeled as a graph in which pixels or voxels constitute vertices and adjacency relationships constitute edges. Such representations allow fundamental topological properties, including connectivity, connected components, paths, boundaries, cycles, holes, and structural continuity, to be studied using discrete mathematical techniques. The present paper examines the application of graph theory to digital topology and image analysis, with particular emphasis on pixel and voxel adjacency, connected-component analysis, image segmentation, boundary detection, skeletonization, region adjacency graphs, medical image analysis, and topological feature extraction. The paper adopts a theoretical and analytical research methodology and integrates classical graph-based digital topology with contemporary developments in computational topology and topological data analysis. Particular attention is given to 4- and 8-adjacency in two-dimensional images and 6-, 18-, and 26-adjacency in three-dimensional images. The study also examines the integration of graph structures with persistent homology and machine-learning techniques. It is observed that graph-based representations transform complex spatial relationships into computationally manageable discrete structures while preserving selected connectivity information. Recent advances further demonstrate that topological features can complement conventional pixel-intensity-based and deep-learning approaches in image classification and segmentation. The paper concludes that the integration of graph theory, digital topology, persistent homology, and artificial intelligence provides a promising mathematical framework for structurally informed image analysis.