In software system design, one of the purposes of diagrammatic modeling is to explain something (e.g., data tables) to others. Very often, syntax of diagrams is specified while the intended meaning of diagrammatic constructs remains intuitive and approximate. Conceptual modeling has been developed to capture concepts and their interactions with each other in the intended domain and to represent structural and behavioral features of the modeled system. This paper is a venture into diagrammatic approaches to the semantics of modeling notations, with a focus on data and graph semantics. The first decade of the new millennium has seen several new world-changing businesses spring to life (e.g., Google and Twitter), that have put connected data at the center of their trade. Harnessing such data requires significant effort and expertise, and it quickly becomes prohibitively expensive. One solution involves building graph-based data models, which is a challenging problem. In many applications, the utilized software is managing not just objects as well as isolated and discrete data items but also the connections between them. Data semantics is a key ingredient to construct a model that explicitly describes the relationships between data objects. In this paper, we claim that current ad hoc graphs that attempt to provide semantics to data structures (e.g., relational tables and tabular SQL) are problematic. These graphs mix static abstract concepts with dynamic specification of objects (particulars). Such a claim is supported by analysis that applies the thinging machine (TM) model to provide diagrammatic representations of data (e.g., Neo4J graphs). The study s results show that to take advantage of graph algorithms and simultaneously achieve appropriate data semantics, the data graphs should be developed as simplified forms of TM.