In the real world, structured data are increasingly represented by graphs. In general, the applications concern the most varied fields, and the data need to be represented in terms of local and spatial connections. In this scenario, the goal is to provide a structure for the representation of a digital image, called the Attributed Relational SIFT-based Regions Graph (ARSRG), previously introduced. ARSRG has not been described in detail, and for this purpose, it is important to explore unknown aspects. In this regard, the goal is twofold: first, to provide a basic theory, which presents formal definitions, not yet specified above, clarifying its structural configuration; second, experimental, which provides key elements about adaptability and flexibility to different applications. The combination of the theoretical and experimental vision highlights how the ARSRG is adaptable to the representation of the images including various contents.