# Glossary

• The adjacency matrix of a graph is the n x n matrix, where A(i,j) is 1 if (i,j) are connected and zero else.
• The characteristic length of a graph is the average distance between two different points. The median length is the median of the distances between to other points.
• A graph is connected if any two points can be connected with a path x(1),x(2),..,x(n) with (x(k),x(k+1)) in E.
• The diameter of a network is the maximal distance between two nodes.
• A finite simple graph G=(V,E) is a graph without multiple connections and without loops. The set V is the set of vertices and the set E is the set of edges.
• Given a subset Y of the vertex set V of a graph G, it generates a subgraph (Y,W), where W consists of all pairs (x,y) with (x,y) in V.
• The degree d(x) of a vertex is the number of vertices in the unit sphere S(x). The average degree is 2|E|/|V|.
• The Laplacian of G is the matrix D-A, where D is the diagonal matrix containing the vertex degrees.
• The local clustering coefficient of a vertex is the number of edges in the unit sphere divided by B(n(x),2), where n(x) is the number of vertices in the unit sphere. The mean clustering coefficient is the average of all the local clustering coefficient.
• The sphere of a vertex x is the sub graph generated by all the vertices attached to x.
• The vertex distribution of a graph gives the probability distribution of the degree function d(x).