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.