## 1 Introduction: The Use of Simple Graphs in the Description of the Fundamental Entinties
The connection between two nodes (vertices) is non-reflective (a node cannot connect to itself) and commutative (if node F\({}_{1}\) is connected to node F\({}_{2}\), it necessarily follows that F\({}_{2}\) is connected to F\({}_{1}\)), but not necessarily transitive (if F\({}_{1}\) is connected to F\({}_{2}\) and F\({}_{2}\) is connected to F\({}_{3}\), it does not mean that F\({}_{1}\) and F\({}_{3}\) are connected). 3. Two nodes (vertices) can be connected by not more than one connection. 4. If two nodes (vertices), F\({}_{1}\) and F\({}_{2}\), are connected, then there must exist an additional third node (vertex), F\({}_{3}\), connected to F\({}_{1}\) and F\({}_{2}\). F\({}_{3}\) will be denoted the \"mediator\" between the nodes F\({}_{1}\) and F\({}_{2}\). 5. It is required that every two nodes (vertices) are different (the meaning of the word \"different\" is to be explained below) in terms of their connections to the other nodes (vertices). In this paper, we show through mathematical proof that there is no ontological structure with less than seven nodes. We also show through construction that there are at least two distinct ontological structures with seven nodes. In addition, we show how the connections (edges) of those graphs can be mapped into the elementary particles (basic fermions and bosons) and how the properties of those connections (elementary particles) can be derived. Thus, the information contained in the above set of axioms is equivalent to the information of the required particles (fundamental fermions and gauge bosons) in the standard model of elementary particles, as we show below. The fifth axiom (axiom e) requires criteria to distinguish between two nodes (vertices) in the ontological structure (OS) graph by their connections to other nodes (vertices). The valency (number of connections) alone cannot serve to uniquely distinguish between two nodes (vertices) because, according to the equivalence theorem, every simple graph has at least two vertices of the same valency, so that axiom e will not be satisfied. An additional criterion, which might serve the purpose of distinction between two nodes (vertices), is given below:  
**Definition 1**. : _The significance set (SS) of vertex \(P\) is the set of all the valences of the nodes connected to \(P\)._  
Figure 1 is given as an example. Let us look at SS(P\({}_{1}\)) =2,3,3,5, since P\({}_{1}\) is connected to P\({}_{2}\), P\({}_{4}\), P\({}_{5}\), and P\({}_{6}\); and the valences of these nodes (vertices), which are given in parentheses in the figure, are, respectively, 3, 3, 2, and 5. By the same method, we derive SS (P\({}_{5}\)) = 4,5. **Definition 2**. : _The vertices \(P_{1}\) and \(P_{2}\) are considered equal; \(P_{1}\) = \(P_{2}\) if, and only if, they have the same valency and SS (\(P_{1}\)) = SS (\(P_{2}\))._  
For example, in Figure 1, P\({}_{3}\)\(\
eq\) P\({}_{2}\). Both vertices have the same number of connections, 3, but they have different significance sets. SS (P\({}_{2}\)) = 3,4,5; and SS (P\({}_{3}\)) = 3,3,5. **Definition 3**. : _A simple, undirected graph respecting all five axioms, a to \(e\), is to be named ontological structure (OS)._  
In the OS graph, every node corresponds to an FE, and every node connection (edge) corresponds to a binary combination of FEs which will be mapped to an elementary particle. The _order_ of an OS is the number of nodes in the graph. Since each node corresponds to an FE, it is expected that the simplest description of physical reality will correspond to the OS with the lowest number of nodes.