# A General Computational Approach for Counting Labeled Graphs
## 2 Terminology  
We represent a graph, \(G=(V,E)\), as an adjacency matrix with dimensions equal to the size of set \(V\). Therefore, \(G\) has dimensions \(|V|\times|V|\), where \(|V|\) denotes the size of set \(V\). Let \(n\) and \(m\) represent the number of vertices in \(G\), i.e., \(n=|V|\) and number of edges, i.e., \(m=|E|\), respectively. Let \(\{v_{1},\ldots,v_{n}\}\) denote the vertices in set \(V\), which are labeled (arbitrarily) but enables them to be distinguishable from one another. Let \((v_{i},v_{j})\) denote an edge between \(v_{i}\) and \(v_{j}\). Let \(G[i,j]=1\) indicate that there is an edge between \(v_{i}\) and \(v_{j}\), where \(v_{i},v_{j}\in\{v_{1},\ldots,v_{n}\}\), while \(G[i,j]=0\) indicates that there is no edge. Denote the neighbors of \(v_{i}\) as \(\eta(v_{i})\), i.e., \(\eta(v_{i})=\{v_{j}:G[i,j]=1\}\). Let \(\mathcal{G}_{n}\) be the set of all simple labeled graphs with \(n\) vertices.  
Let \(\phi_{1}\) denoted an algebraic map from a graph \(G\) to its number of edges, i.e., \(\phi_{1}(G)=m\). The degree of vertex \(v_{i}\), denoted as \(d_{i}(G)\), is the number of edges the vertex has with other vertices in \(V\); therefore \(d_{i}(G)=\sum_{j}G[i,j]\). Let \(d(G)=(d_{1}(G),\ldots,d_{n}(G))\) represent the vector of degrees for nodes in set \(V\), commonly referred to as a degree sequence. The degree distribution, denoted as \(D(G)\), is a vector representing the number of these degrees over all vertices in set \(V\); the \(k\)th entry represents the number of vertices with degree \(k\), i.e., \(D_{k}(G)=\sum_{i=1}^{n}I_{\{d_{i}(G)=k\}}\). Let \(\phi_{2}\) and \(\phi_{2a}\) denote the mapping from a graph to its degree distribution, i.e., \(\phi_{2}(G)=D(G)\), and degree sequence, i.e., \(\phi_{2a}(G)=d(G)\), respectively.  
Let \(m_{i}(G)\) represent a discrete classification for vertex \(v_{i}\) in graph \(G\); we denote the number of distinct classifications as \(q\). Let \(m(G)=(m_{1}(G),\ldots,m_{n}(G))\) be a vector containing the characteristics of all vertices. The classification distribution, denoted as \(M(G)\), is a vector representing the number of individuals with these classification over all vertices; the \(k\)th entry represents the number of vertices with classification \(k\), i.e., \(M_{k}(G)=\sum_{i=1}^{n}I_{\{m_{i}(G)=k\}}\). Let \(MM(G)\) be a \(q\times q\) symmetric matrix representing the mixing by classification of graph \(G\); we refer to \(MM(G)\) as a classification mixing matrix. The entry \(MM_{k,l}(G)\) is the total number of edges between a vertex with classification \(k\) and vertex with classification \(l\). Let \(\phi_{3}\) denote the mapping from a graph to its classification mixing matrix. Let \(DMM(G)\) be a particular mixing matrix where the classification represents vertex degrees. Therefore, the entry \(DMM_{k,l}(G)\) is the total number of edges between vertices of degree \(k\) and \(l\). Let \(\phi_{4}\) denote the mapping from a graph to its degree mixing matrix.  
Denote the inverse images associated with a map \(\phi_{i}\) as \(c_{\phi_{i}}(x)=\{G:\phi_{i}(G)=x,G\in\mathcal{G}_{n}\}\). These inverse images of singleton sets have been referred to as fibers in algebraic statistics literature [11]. The graph enumeration problem is calculating the size of a fiber,denoted as \(|c_{\phi_{i}}(x)|\), which represents the number of graphs where the graph property associated with \(\phi_{i}\) equals \(x\); this quantity has been referred to as a volume factor [12]. We refer to \(x\) as a graphical value associated with \(\phi_{i}\) if \(|c_{\phi_{i}}(x)|\geq 1\).