Summary Van Steen: Graph Theory and Complex Networks

VAN STEEN: GRAPH THEORY AND COMPLEX NETWORKS

A graph G consists of a collection V of vertices and a collection of edges E  G = (V,E).
Each edge e ∈ E joins two vertices, which are called its endpoints  the two vertices are adjacent, with the edge
having an incident with the vertices.

Simple graph: a graph that does not have loops (an edge connects just one vertex with itself) or multiple edges
(multiple edges that have the same endpoints).
Empty graph: a graph containing no vertices nor edges.
Complete graph: a simple graph with each vertex being adjacent to every other vertex  Kn , with n = total vertices.

Complement of a graph G: graph obtained from G by removing all its edges and joining the vertices that were not
adjacent in G.

The neighborset N(v) of v: N(v) = {w ∈ V(G) | v ≠ w, ꓱ e ∈ E(G) : e = <u,v> }  the set of vertices w in G, with w not
equal to v, such that there exists an edge e that joins v and w.

Degree of a vertex: the number of edges incident with the vertex.

𝑆𝑢𝑚 𝑜𝑓 𝑣𝑒𝑟𝑡𝑒𝑥 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 = ∑ 𝛿(v) = 2 ∙ |𝐸(𝐺)|
𝑣 ∈V(G)
For any graph, the number of vertices with an odd degree is even.

Subgraph: a graph is a subgraph of another graph if it consists of a subset of edges and vertices of the other graph 
graph H is a subgraph of G if V(H) ⊆ V(G) and E(H) ⊆ E(G) such that for all e ∈ E(H) with e = <u,v>, we have that u, v ∈
V(H). When H is a subgraph of G, we write H ⊆ G.
Induced subgraph: subgraph induced by a subset of vertices and edges linking those edges from the original graph 
subgraph induced by V* has vertex set V* and edge set E* defined by E* = {e ∈ E(G) | e = <u,v> with u,v ∈ V* }

Line graph: a graph that represents every edge by a vertex, and joining two vertices if the edges are incident with the
same vertex in the main graph.

Representing graphs:
 Adjacency matrix: a table displaying the number of edges joining vertices. The sum of the values in a row is
equal to the degree of the vertex.
 Incidence matrix: a table counting the amount of times edges incident with vertices, with the vertices on the
y-axis and the edges on the x-axis  if there are no loops in the graph, the values in the matrix will always be
0 or 1.
The sum of all values in a row is equal to the degree of the vertex.
 Edge list: listing all edges of a graph by specifying for each edge which vertices it is incident with. This is more
commonly used for larger graphs, as the amount of data needed is far less than when we’d store the matrixes.

Graph isomorphism: two graphs are isomorphic if both graphs have the same amount of vertices and their edges are
also linked to the same vertices.

A graph is connected if all pairs of vertices are connected: between the vertices a path exists.

Component: a subgraph that is not contained in a connected subgraph of the main graph with more vertices or edges
 maximal, connected subgraph. 𝜔(G) is the total amount of edges of graph G.

Robustness: how well a network stays together when we remove vertices or edges.

Vertex cut: set of vertices that needs to be removed to make a connected graph disintegrate into several components
 disconnect.
Cut vertex: a single vertex that needs to be removed to make a connected graph disintegrate into several
components.