## 3 Characteristics of OS Graphs  
An interesting problem is the construction of the graphs of all possible OSs of a given order, n. The goal of obtaining a general solution to this problem has not been achieved yet, and we hope that experts in the enumeration of graphs satisfying specified conditions [15] will be able to help in this endeavor. We believe that the following concepts and theorems have some potential in resolving this problem. In the subsequent theorems and proofs, we use the following notations:  
V(P) denotes the valency of the vertex P.  
O(n)--OS graph of order n satisfying axioms a to e.  
The multiplicity of k, m\({}_{k}\), is the number of vertices having valency k. (See Definition 4). The index I of O(n) is the number of vertices in the OS having different valency. (See Definition 5). **Theorem 1**. : _Let the vertices of O(n) be \(P_{1}\), \(P_{2}\),..., \(P_{n}\). Let \(K_{i}=V(P_{j})\) (\(1\leq i\leq n\)). If \(max(K_{i})<n\) - 1, then it is possible to enlarge the O(n) to O(n + 1) by adding the vertex \(P_{n+1}\)* and by connecting it to all the n vertices of O(n)._  
Proof. : Let us denote by P\({}_{1}\)*, P\({}_{2}\)*,..., P\({}_{n}\)* the vertices P\({}_{1}\),..., P\({}_{n}\) when they are connected to P\({}_{n+1}\)*. It is clear that V(P\({}_{1}\)*) =V(P\({}_{1}\)) + 1 for \(1\leq i\leq n\). We have to show that the set of the vertices P\({}_{1}\)*, P\({}_{2}\)*,...P\({}_{n+1}\)* satisfies the basic axioms, a to e. The proof of the validity of axioms a to c is trivial. Since P\({}_{n+1}\)* is connected to all the n vertices, axiom d is satisfied for P\({}_{n+1}\)*. The fact that the set P\({}_{1}\),..., P\({}_{n}\) is an OS guarantees that axiom d is satisfied also for the vertices P\({}_{1}\)*,..., P\({}_{n}\)*. Now, let us turn to axiom e. It is clear that V(P\({}_{n+1}\)*) \(\
eq\) V(P\({}_{1}\)*) for every i \(\leq n\). Since we assumed that \(max\{V(P_{i})\}<n-1\), then \(max\{V(P_{i})\}<n\). It is easy to show that, for every two vertices, P\({}_{1}\)* and P\({}_{j}\)* in O(n + 1) (\(1\leq i\
eq j\leq n\)) P\({}_{1}\)*\(\
eq\)P\({}_{j}\)* (because P\({}_{i}\)\(\
eq\) P\({}_{j}\) in O(n). ), so that axiom e is satisfied for all the vertices of O(n + 1). \(\square\)  
As a result of Theorem 1, we obtain the following theorem: \begin{table}
\begin{tabular}{c c} \hline
**Terms in Apriorics** & **Terms in Graph Theory** \\ \hline Ontological structure, OS & Undirected simple graph satisfying axioms a to e \\ \hline Fundamental entity (FE) & Vertex \\ \hline B compound (BC) & Edge \\ \hline Order of OS & Order of the graph \\ \hline Number of BCs & Size of the graph \\ \hline Significance set (SS) & See Definition 1 \\ \hline Equality between vertices & See Definition 2 \\ \hline \end{tabular}
\end{table} 
Table 5: Comparison of the terms used in apriorics and the corresponding terms in graph theory. **Theorem 2**. : _In \(O(n)\), if max{V(P_{i})} = \(n-1\), (\(1\leq i\leq n\)), then by adding the vertex \({P_{n+1}}^{*}\) and by connecting it to all the other vertices, we do not obtain an OS._  
Proof. : Suppose that V(P\({}_{k}\)) = n - 1; then, V(P\({}_{k}\)*) = n = V(P\({}_{n+1}\)*). Also, since P\({}_{k}\)* and P\({}_{n+1}\)* are connected to all the other vertices, they have the same SS, so that P\({}_{k}\)* = P\({}_{n+1}\)*, and axiom e is not satisfied. \(\square\)  
A magnitude of some use in the analysis of OS is the multiplicity of a valency, V, defined as follows. **Definition 4**. : _In \(O(n)\), the vertices having the same valency shall be called equivalent vertices. The number of vertices, \(m_{k}\), having \(V\) = \(k\) shall be called the multiplicity of \(k\)._  
**Example 1**. : _The multiplicities of the vertices of the OS \(O_{1}\)(7) and \(O_{2}\)(7) discussed in the previous section are given in Table 6._  
**Definition 5**. : _The index of \(O(n)\), \(I\), is the number of vertices in the OS with a different valency._  
**Example 2**. : _In \(O_{1}\)(7), \(I\) = 4 since there are 4 different equivalent subsets corresponding to the valencies 2, 3, 4, and 5.