Graphs
Definition
A simple graph G= (V, E) consists of vertices, V, anonempty set of vertices, and E, a set of unorderedpairs of distinct elements of V called edges.
•- no arrows
•- no loops
•- can't have multiple edges joining vertices
A simple graph
Definition
A multigraph G=(V, E) consists of a set V of vertices,a set E of edges, and a function f from E to {{u, v} |u, v V, u v}. The edges e1 and e2 are calledmultiple or parallel edges if f(e1) = f(e2).
A multigraph
No loop is allowed.
• Multiple edges are allowed.
Definition
A pseudograph G = (V, E) consists of a set V ofvertices, a set E of edges, and a function f from E to{{u, v} | u, v V}. An edge is a loop if f(e) = {u, v} forsome v V.
Definition
◦A directed graph (V, E) consists of a set of a set ofvertices V and a set of edges E that are orderedpairs of elements of V.
Loops, ordered pairs orthe same element, areallowed.
Multiple edges in thesame direction betweentwo vertices are notallowed.
A directed graph
Definition
◦A directed multigraph G = (V, E) consists of a set Vof vertices, a set E of edges, and a function f from Eto {(u, v) | u, v V}. The edges e1 and e2 aremultiple edges if f(e1) = f(e2).
•Loops, ordered pairs orthe same element, areallowed.
Multiple edges in thesame direction betweentwo vertices are allowed.
A directed multigraph
Summary
Type
Edges
Multiple edgesallowed?
Loopsallowed?
Simplegraph
Undirected
No
No
Multigraph
Undirected
Yes
No
Pseudograph
Undirected
Yes
Yes
Directedgraph
Directed
No
Yes
Directedmultigraph
Directed
Yes
Yes
There will be an edge (a, b) from team a toteam b, if team a beats team b.
•Adjacent:
•Two vertices u and v in an undirected graph G arecalled adjacent (or neighbours) if {u, v} is an edgeof G.
•If e = {u, v} the edge e is called incident withu and v. The edge e is also said to connect uand v. The vertices u and v are calledendpoints of the edge {u, v}.
15
a
b
c
d
e
e1
e2
e3
e4
e5
e6
•Vertex a is adjacent to b because there is an edge e1that connects vertices a and b.
•Edge e4 is incident with vertices a and d.
•Edge e4 connect vertices a and d.
•Edge e6 connect vertices e and e.
•The degree of a vertex in an undirectedgraph is the number of edges incident with it,except that a loop at a vertex contributestwice to the degree of that vertex.
•The degree of the vertex v is denoted bydeg(v).
18
What are the degrees of the vertices in thefollowing graph?
•
•
•
•
•
•
•
b
a
c
f
e
d
g
deg(a) =
deg(b) =
deg(c) =
deg(d) =
deg(e) =
deg(f) =
deg(g) =
2
4
4
3
1
4
0
Vertex of degree zero is calledisolated
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Theorem:
Suppose the vertices of graph G are v1, v2, …,vn, where n is a non-negative integer, thenthe total degree of G
=deg(v1) + deg(v2) + …. + deg(vn)
= 2 (the number of edges of G).
Total number of edges: 4
Total degree: 1 + 1 + 1 + 5 = 8.
Handshaking theorem: 2 x 4 = 8.
21
How many edges are there in a graph withseven vertices each of degree four?
•The sum of the degrees of the vertices is
4x7 = 28.
2e = 28
e = 14.
Three methods
1.Adjacency lists
2.Adjacency matrices
3.Incidence matrices
Adjacency lists
b
a
c
d
e
A simplegraph
UNABLE TO REPRESENT MULTIPLE-EDGES.
A directed graph
Adjacency matrices
Suppose that G=(V, E) is a simple graph where |V|=n.Suppose that the vertices of G are listed arbitraryas v1,v2,..., vn. The adjacency matrix A of G is a n×nzero-one matrix with 1 as its (i, j)th entry when viand vj are adjacent, and 0 as its entry when theyare not adjacent.
a
b
c
d
G
Adjacencymatrix for G
Representing multigraph or pseudograph
0 1 0 0 1
0 1 0 0 0
1 0 0 1 0
0 0 1 0 1
0 0 0 0 0
a b c d e
a
b
c
d
e
Adjacency Matrix
Incidence Matrices
Let G=(V, E) be an undirected graph. Supposethat v1, v2, ..., vn are the vertices and e1, e2, ...,em are the edges G. The incidence matrix of G isa nx m matrix M=[mij], where
Incidence Matrix
Definition
Let n be a nonnegative integer and G a directedmultigraph. A path of length n from u to v in Gis a sequence of edges e1, e2, ..., en of G suchthat f(e1)=(x0, x1), f(e2)=(x1, x2), f(en)=(xn-1, xn),where x0=u and xn=v.
Definition
When there are non multiple edges in thedirected graph, this path is denoted by itsvertex sequence x0, x1, x2, ..., xn.
A path of length greater than zero that beginsand ends at the same vertex is called a circuitor cycle.
A path or circuit is simple if it does not containthe same edge more than once.
a, d, c, f, e is a simple path of length 4 since {a, d}, {d, c},{c, f}, and {f, e} are all edges and no repeated edge.
b, c, f, e, b is a circuit of length 4 since this path begins andends at b.
The path a, b, e, d, a, b is of length 5, is not simple since itcontains the edge {a, b} twice.
Is a, d, e, a, b a simple path?
Paths from v0 to v7
1. v0 v1 v2 v5 v7
2. v0 v1 v4 v5 v4 v5 v7
3. v0 v3 v4 v6 v7
Which path(s) is (are)simple?
Definition
An undirected graph is called connected if thereis a path between every pair of distinct verticesof the graph.
G is connected,whereas H is not.
Theorem
There is a simple path between every pair of distinctvertices of a connected undirected graph.
Definition
A directed graph is strongly connected if there isa path from a to b and from b to a whenever aand b are vertices in the graph.
Definition
A directed graph is weakly connected if there isa path between any two vertices in theunderlying undirected graph.
G is strongly connected because there is a pathbetween any two vertices in this directed graph.
The graph H is not strongly connected. There is nodirected path from a to b in this graph. H is weaklyconnected since there is a path between any twovertices in the underlying undirected graph of H.
Questions
Can we travel along the edges of graph startingat a vertex and returning to it by traversingeach edge of the graph exactly once?
Can we travel along the edges of a graphstarting at a vertex and returning to it whilevisiting each vertex of the graph exactly once?
Definition
An Euler circuit in a graph G is a simple circuitcontaining every edge of G.
An Euler path in G is a simple path containingevery edge in G.
Note: Both in Euler path and Euler circuit, each edgecannot be repeated more than once.
If a graph has Euler circuit then it must has Euler path,the opposite could be false.
Example
•Theorem 1
•A connected multigraph has an Euler circuit if andonly if each of its vertices has even degree.
•Theorem 2
•A connected multigraph has an Euler path but notan Euler circuit if and only if it has exactly twovertices of odd degree.
For graph G1, the degree for each vertex in thisgraph is even, hence this graph contain Euler circuit.
For graph G2, the degree for vertex e, a are odd, sothere is no Euler circuit.
G1
G2
G1 contains exactly two vertices of odd degree, b and d.Hence it has an Euler path that must have b and d as its endpoints. E.g.:
b, c, d, a, b, d.
G2 also contains exactly two vertices of odd degree, d and b.One of the Euler path is b, a, g, b, c, g, f, c, f, e, d.
Definition
•A path x0, x1, ..., xn-1, xn in the graph G = (V, E) iscalled a Hamilton path if V={x0, x1, ..., xn-1, xn} andxixj for 0 ijn
•A circuit x0, x1, ..., xn-1, xn , x0(with n > 1) in a graphG = (V,E) is called a Hamilton circuit
◦if x0, x1, ..., xn-1 Hamilton path.
Graph G1 has a Hamilton circuit: a, b, c, d, e, a.
There is no Hamilton circuit in G2 because edge (a, b)will be always use twice. E.g.: d, c, b, a, b, d. G2 hasHamilton path, a, b, c, d.
•A graph with a vertex of degree 1 cannothave a Hamilton circuit.
•If each vertex in a graph is adjacent to everyother vertex there is always a Hamiltoncircuit.