Discrete Math Notes:
Chapter 6: Graphs and Trees
6.1 Introduction to graphs
undirected graph, edges are unordered pairs of vertices, which is useful for modeling relationships that are symmetric
finite, the vertex set is finite
vertex, a single element of V is called
Parallel edges, are multiple edges between the same pair of vertices. Imagine a graph whose vertex set is a set of cities and whose edges are roads connecting pairs of cities.
simple graph, a graph does not have parallel edges or self-loops
graph
adjacent, an edge between two vertices
endpoints, vertices b and e
incident the edge {b, e}
neighbor, A vertex c
degree, a vertex is the number of neighbors
total degree, a graph is the sum of the degrees of all of the vertices
regular graph, all the vertices have the same degree
d-regular graph, all the vertices have degree d.
subgraph,
clique,
complete graph, Kn is called on n vertices
overview:
Example of Graphs
Figure 6.1.4: A graph with parallel edges and a self-loop.
Figure 6.1.5:Undirected graph example.
The edges of an undirected graph are unordered pairs of vertices, which is handy for describing symmetric interactions. An undirected graph, for example, could be used to model sibling connections in a family. Sibling relationships are symmetrical, unlike parent-child interactions, in which the two people play separate roles. Two people are mutual siblings if neither of them is the other's sibling. The vertex set of a graph is finite if it is finite. Only finite graphs will be discussed in this section. A vertex is a single element of V that is commonly represented graphically by a dot with a label. A self-loop, or edge between a vertex and itself, can exist in a graph. Between vertices a and b, there are two parallel edges in the graph below. A graph is said to be simple if it does not feature parallel edges or self-loops. An undirected graph in this material is presumed to be a simple graph unless otherwise specified.
Discrete Math Notes:
Chapter 6: Graphs and Trees
6.3 Graph isomorphism
Isomorphic, a correspondence between the vertex sets of each graph such that there is an edge between two vertices of one graph if and only if there is an edge between the corresponding vertices of the second graph.
preserved under isomorphism, whenever two graphs are isomorphic, one graph has the property if and only if the other graph also has the property.
degree sequence, a list of the degrees of all of the vertices in non-increasing order.
Theorem 6.3.1: Vertex degree preserved under isomorphism, Let's assume that two graphs, G and G'. Let f be an isomorphism from G to G'. For each vertex v in G, the degree of vertex v in G is equal to the degree of vertex f(v) in G'.
Overview:
The two graphs' drawings indicate a connection between the two sets of vertices, making it simple to check that each edge in the left graph corresponds to an edge