graphs
DESCRIPTION
Graphs. Simple graph G=(V,E) V = V(G) ={1,2,3,4} – vertices E = E(G) = {a,b,c,d,e} – edges Edge a has end-vertices 1 and 2. Vertices 1 and 2 are adjacent: 1 ~ 2. a. 1. 2. c. b. d. e. 3. 4. Simple Graph. - PowerPoint PPT PresentationTRANSCRIPT
Graphs
• Simple graph G=(V,E)• V = V(G) ={1,2,3,4} –
vertices• E = E(G) = {a,b,c,d,e}
– edges• Edge a has end-
vertices 1 and 2. Vertices 1 and 2 are adjacent: 1 ~ 2.
1
3 4
2a
e
c bd
Simple Graph
• Definition: Graph X is composed of the set of vertices V(X) endowed with irreflexive symmetric relation ~ (adjacency). An unoredered pair of adjacent vertices uv = vu forms an edge. The set of edges is denoted by E(X). Sometimes we write X = (V,E) or X(V,E).
Families of Graphs
Cycle Cn on n vertices.
V – vertices of a regular n-gon
E – edges
• |V|=n• |E|=n
1
3 4
2
C4
Small Cycles
• Some cycles as drawn by VEGA.
• It makes sense to define cylces C1 (a loop) and C2 (parallel edges), that are NOT simple.
C3 C4
C5 C6C1 C2
Path Pn on n vertices.
V – vertices of polygonal line.
E – segments. The endpoints of the
polygonal line are called the endpoint of the path.
For instance, 1 and 4 are the endpoints of the path on the left.
• |V|=n• |E|=n-1P4
1
3 4
2
1 3 42
Complete graph on n vertices Kn.
V – vertices of a regular n-gon
E – edges and diagonals.
• |V|=n• |E|=n(n-1)/2
1
3 4
2
K4
Complete Bipartite Graph on n+m vertices Kn,m.
V = U1 U2 , U1 Å U2 = ;
|U1| = m, |U2 | = n.
E = U1 U2
• |V|=n + m
• |E|=n m
1
3 4
2
K2,2
Metric Space
• Space V, with mapping d (distance):
• d:V V R with the following properties:
• d(u,v) ¸ 0, d(u,v) = 0, iff u = v.
• d(u,v) = d(v,u)
• d(u,v) · d(u,w) + d(w,v)
• is called a metric space with distance d.
Example: Hamming Distance
{0,1}n is a metric space if distance between u and v is the number of components in which the two vectors differ.
– E.g. d([0,0,0,1,0,1],[1,1,0,1,1,1]) = 3. – d is called the Hamming distance.
Hypercube Qn.
• Hypercube of dimension d is the graph Qn, with:
• V(Qn) = {0,1}n.
• u ~ v, if d(u,v) = 1.
• |V(Qn)| = 2n
• |E(Qn )|= n 2n-1
Q1Q2 Q3
Q4Q5
Vertex Valence
• G = (V,E) • V(G) ={1,2,3,4} • E(G) = {a,b,c,d,e}• Number of edges incident with
vertex v is called the valence or degree of v: deg(v).
• deg(1) = deg(4) = 3, deg(2) = deg(3) = 2.
• Vertex of valence 1 is called a leaf, vertrex of valence 0 is isolated.
• (G) – minimal valence.• (G) – maximal valence.
4
1
3
2a
e
c bd
Regular GraphsGraph G is regular (of
valence k), if G) = G) = k.
Zgled:• Regular graphs: Kn, Cn, Kn,n
• Nonregular graphs: Pn, n > 2, Kn,m, n m.
1-valent and 2-valent graphs have simple structure. Trivalent graphs have special name: cubic graphs. (See example on the left)
Girth
• Girth g(G) of graph G is the number of vertices of the shortest cycle in G. If G has no cycles, its girth is infinite.
Cages
• Graph G is a g-cage, if the following holds:
1. Trivalent
2. Has girth g
3. Has the least number of vertices among the graphs satisfying 1 and 2.
Exercises 01
• N1. Deterimine the 3-cage.
• N2. Determine the 4-cage.
• N3. Determine the 5-cage.
• N4. Determine the 6-cage.
The Petersen Graph and its Generalizations G(n,k)
• Petersen graph G(5,2) is an example of a generalized Petersen graph G(n,k).
• V(G(n,k)) consists of • ui, vi, i = 1,2, ..., n.Edges:• ui ~ ui+1
• ui ~ vi
• vi ~ vi+k
(Warning! Addition mod n)
Examples of Generalized Petersen graphs
• G(10,2) Dodecahedron• G(10,3) Desargues
graph.• G(8,3) Möbius-Kantor
graph.• G(6,2) Dürer graph.
Incidence Matrix M(G).
• To G=(V,E) we associate a rectangle matrix M=M(G) with |V| rows and |E| columns:
otherwise0 ...
Mv,e =
v is the endpoint of e 1 ...
{
Incidence Matrix - Example
• G=(V,E) • VG ={1,2,3,4} • EG = {a,b,c,d,e}
MG =
1
3 4
2a
e
c bd a b c d e
1 1 0 1 1 0
2 1 1 0 0 0
3 0 0 0 1 1
4 0 1 1 0 1
Handshaking Lemma
• In each graph G=(V,E) : • 2 |E(G)| = v 2 V(G) deg(v),
• The proof uses the so-called bookkeepers rule in the incidence matrix of graph G.
Graph Invariant
• It is well-known that we associate numbers to mathematical objects in various ways. For instance: Determinant is assicated to a matrix, degree is associated to a polynomial, dimension is associated to a space, length is associated to a vector, etc.
• There are several numbers that can be associated with a graph. Such a number is usually called graph invariant. One may argue that the main topic of graph theory is the study of graph invariants.
• In addition to numbers other objects may be graph invariants.
Isomorphisms and Graph Invariants
Isomorphism G) = H is a bijective mapping:
• : V(G) ! V(H).that preserves adjacency: • u ~ v if and only if (u)~(v).Graph invariant is a property, (usually a
number), that is preserved under an isomorphism.
Isomorphism - Exercises
• N1. Determine an isomorphism between graphs A and B.
• N2. Determine an isomorphism between graphs C and D.
A B
C D
Adjacency Matrix A(G).
• To each graph G=(V,E) with V={1,2,3,...,n} we can associate the adjacency matrix A=A(G) as follows:
sicer0 ...
Ai,j =
i ~ j 1 ...
{
Adjacency Matrix - Example
• G=(V,E) • VG ={1,2,3,4} • EG = {a,b,c,d,e}
AG =
1
3 4
2a
e
c bd 0 1 1 1
1 0 0 1
1 0 0 1
1 1 1 0
Adjacency Matrix is Not an Invariant
• Adajcency matrix is not an invariant. It depends on the numbering of vertices.
• Incidence matrix is not an invariant
Some Graph Invariants
• |V(G)| = number of vertices
• |E(G)| = number of edges
• G) = minimal valence.
• G) = maximal valence
Invariants - Example
• |V(G)| = 4 • |E(G)| = 5• G) = 2• G) = 3
1
3 4
2a
e
c bd
Trees
• A tree is a connected graph with no cycles
• There are several characterizations of tree, such as:
• A tree is a connected graph with n vertices and n-1 edges.
• A tree is a connected graph that is no longer connected after removal of any edge.
Disjoint Union of Sets
• Let A and B be sets. By A t B we denote the disjoiont union of A and B. If A Å B = ;, then A t B is simply the union of the two sets. Otherwise we defne formally A t B = A £ {0} [ B £ {1}.
Disjoint Union of Graphs
• Let G’ and G” be graphs. By G’ t G” we denote the disjoiont union of graphs G’ and G”. This means
• V(G’ t G”) := V(G’) t V(G”) and
• E(G’ t G”) := E(G’) t E(G”).
The Empty Graph
• Empty graph has no vertices and no edges.
Connectivity in Graphs - Theory
• Graph G is connected, if and only if it cannot be written as a disjoint union of two non-empty graphs.
Connectivity of Graphs - Practice
• Graph is connected, if we grab and shake the “model” made of balls and strings, and nothing falls down the earth. (No knotting of strings is permitted!)
Equivalence Relation .
• Let G be a graph. On V(G) define as follows: For any u,v 2 V(G) let u v, if and only if there exists a subgraph, isomorphic to a path that has the endponts u and v.
• Proposition. is an equivlanece relation on V(G).
• Proof. Obviously reflexive and symmetric. Proof of transitivity – Homework.
Path Connectivity of Graphs
• G is connected by paths, if the equivalence relation has a single equivalence class.
Homework
• H1: Prove that the relation is transitive.
• H2: Prove that for finite graphs the notions of connectedness and path connectedness coincide.