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Basic Notions

on Graphs

Presented by

Joe RyanSchool of Electrical Engineering

and Computer Science

University of Newcastle, Australia

Matrix Representations

Operators and Trees



2

Incidence matricesLet G be a graph without loops, with n vertices labelled

1,2,…,n , and m edges labelled 1,2,3,…,m . Theincidence matrix I (G ) of G is the n xm matrix in whichthe entry in row i and column j is

1 if the vertex i is incident with the edge j , and0 otherwise.



4

Incidence matrices

Problem Draw the graph represented by each of thefollowing incidence matrices.



5

Incidence matricesLet D be a digraph without loops, with n vertices labelled

1,2,…,n , and m arcs labelled 1,2,3,…,m . Theincidence matrix I (D ) of D is the n xm matrix in which

the entry in row i and column j is1 if the arc j is incident from vertex i ,

-1 if the arc j is incident to vertex i , and

0 otherwise.



6

Incidence matrices

Problem Write down the incidence matrix of each ofthe following digraphs.



8

Null graphsA null graph is a graph with no edges.

The null graph with n vertices is denoted by N n .The graph N n is regular of degree 0.



9

Regular graphsA graph is regular if its vertices all have the same degree.A regular graph is r-regular , or regular of degree r , if the

degree of each vertex is r.



10

Regular graphsExercise: Draw an r -regular graph with 8 vertices

when r = 3,4,5.

Theorem: Let G be an r -regular graph with n vertices. ThenG has nr /2 edges.

Proof. Let G be a graph with n vertices, each of degree r .Then the sum of the degrees is nr . By the Handshaking

Lemma, the number of edges is half of this sum.



11

Regular graphsExercise: Verify that the Theorem holds for each of the

following regular graphs:

Exercise: (a) Prove that there are no 3-regular graphs

with 7 vertices;(b) Prove that, if n and r are both odd, then

there are no r -regular graphs with n

vertices.



12

Cycle graphsA cycle graph is a graph consisting of a single cycle of

vertices and edges.

The cycle graph with n vertices is denoted by C n .The graph C n is regular of degree 2 and has n edges.

Exercise: Draw the graphs K 7, N 7 and C 7.



13

Petersen graphPetersen graph was discovered by Julius Petersen in 1898.

Petersen graph is a 3-regular graph with 10 vertices and

15 edges. It may be drawn in many ways, for example:



14

Platonic graphsPlatonic solids and the corresponding Platonic graphs :



15

CubesCubes : vertices are all binary words of a given length k ,

and two vertices are joined whenever the verticesdiffer in exactly one bit.

k-cube or k-dimensional cube is based on words oflength k ; it is denoted by Q k .



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Bipartite graphsA bipartite graph is a graph whose set of vertices can be

split into 2 subsets A and B in such a way that eachedge of the graph joins a vertex in A and a vertex in B .

Exercise: Prove that in a bipartite graph every cycle has

an even number of edges.



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Complete bipartite graphsA complete bipartite graph is a bipartite graph in which

each vertex in A is joined to each vertex in B byexactly one edge.

K r,s denotes a complete bipartite graph with r vertices in Aand s vertices in B.

Exercise: (a) Draw the graphs K 2,3, K 1,7 and K 4,4. Howmany vertices and edges does each have?

(b) Under what conditions on r and s is K r,s

a regular graph?



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Path graphsA path graph is a tree consisting of a single path through

all its vertices.

Path graph with n vertices is denoted by P n .

The graph P n has n -1 edges and can be obtained from the

cycle graph C n by removing one edge.



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TreesA tree is a connected graph with no cycles.

Note that in a tree there is exactly one path between anytwo vertices.

Exercise: There are 8 unlabelled trees with 5 or fewervertices. Draw them.

Exercise: Explain why every tree is a bipartite graph.

Explain why a tree with n vertices has n -1 edges.



20

Complete graphsA complete graph is a graph in which each vertex is joined

to each of the others by exactly one edge.

The complete graph with n vertices is denoted by K n .The graph K n is regular of degree n -1, and has n (n -1)/2

edges.



21

Complete graphs

Every graph on n vertices is a subgraph of Kn.

|V(G)| = n ⇒ G ⊆ Kn

So we also know that Δ(G) ≤ n-1,|E(G)| ≤ n (n -1)/2 andradius(G) ≥ 1.

And, of course, |V(G)| = n ⇒ G ⊆ Km, m ≥ n.



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Complement of a graphThe complement of a graph G (written G) is a graph on the

same vertex set as G containing all edges not in G.

G G



23

Complement of a graphIf |E(G)| = e, then |E(G)| = n(n-1)/2 - e

G G Kn

+ =



24

Self Complementary Graphs

C5 C5 P4 P4



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Converse of a Digraph For a digraph G, the converse of G is

obtained by simply reversing the directionof the arrows.

A

B

C

DA

B

C

D



26

Self Converse Digraphs

K3 is the same as K3.

C6 is isomorphic to C6.



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Cartesian Products of Graphs1

3

b

2

b3

b2

b1

a c

a3

a2

a1

c3

c2

c1

G1

G2

G1× G

2

The Cartesian Product ofG1 and G2 is the graphobtained by placing acopy of G2 at each vertexof G1 and then joiningcorresponding vertices ofG2 for copies that are

placed at adjacentvertices of G1.



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Duality

Let G be a connected planar graph. Then a dual graph G *is constructed from a plane drawing of G as follows.

Draw one new vertex in each face of the plane drawing:these are the vertices of G *. For each edge e of the planedrawing, draw a line joining the vertices of G * in the faces

on either side of e : these are the edges of G *.



29

Duality

Consider the graph of the cube. If we place a new vertexwithin each face (incl. the infinite face) and join the pairsof new vertices in adjacent faces, we obtain the graph ofthe octahedron and vice versa .



30

DualityProblem Draw the dual of each of the following plane

drawings of planar graphs.

Problem The following diagrams show two different planedrawings of a planar graph. Show that their duals are

not isomorphic.



31

DualityDifferent plane drawings of a planar graph G may give rise

to non-isomorphic dual graphs G *.

If G is a plane drawing of a planar connected graph then sois its dual G *, and so we can construct (G *)*, the dual ofG *.

Note that (G *)* is isomorphic to G .



32

Connectivity

G1 is a tree – removal of any edge disconnects it (all edges are

cut-edges or bridges).G2 cannot be disconnected by removing an edge.but it can be disconnected by removing a vertex (the cut-vertex).

G3 cannot be disconnected by removing an edge or a vertexbut is not as strongly connected as G4.



33

Vertex Connectivity A graph is called k connected if the removal of k

vertices is required to disconnect the graph.

3 connected 2 connected



34

Edge ConnectivityA graph is called k edge connected if the removal of k

edges is required to disconnect the graph.

The above graph is 3-edge connected

Problem: Identify the cut set.



35

ClusteringA graph shows high clustering if neighbours of

points are connected.

This graph is locally connected. All neighbours ofeach point are connected.

Many social network graphs display high clustering



36

ClusteringMost graphs with high clustering have large

diameter.

Small world networks show high clustering buthave small diameter.



37

What is the degree of Facebook?

Stanley Milgram (1967) sent 160 letters

from Omaha, Nebraska to Boston – not by post!

6 degrees of separation



38

Tree structuresA tree is a connected graph that has no cycles.Trees are relatively simple structures but very important formany practical applications.



39

Tree structuresExample of an artificial object that can be modeled as a tree.



40

Tree structuresExample of a conceptual tree: family tree.



41

Tree structuresAnother example of a conceptual tree: hierarchical treerepresenting the responsibilities in a company.



42

Mathematical properties of treesA tree is a connected graph that has no cycles.



43

Mathematical properties of treesProblem Draw the 6 unlabelled trees with 6 vertices.

Each unlabelled tree with n vertices can be obtained from an unlabelled tree with n-1 vertices by adding an edge

joining a new vertex to an existing one.

For example, from

we can obtain



44

Mathematical properties of treesTheorem: Equivalent Definitions of a Tree.Let T be a graph with n vertices. Then the followingstatements are all equivalent.

•T is connected and has no cycles.•T has n -1 edges and has no cycles.•T is connected and has n -1 edges.

•T is connected and the removal of any edgedisconnects T .

•Any two vertices of T are connected by exactly one

path.•T contains no cycles, but the addition of any newedge creates a cycle.

Prove the equivalences.



45

Spanning treesLet G be a connected graph. Then a spanning tree in G

is a subgraph of G that includes every vertex of G andis also a tree.

The number of spanning trees in a graph can be verylarge. For example, the Petersen graph has 2000labelled spanning trees.



46

Spanning treesTwo methods for constructing a spanning tree in a

connected graph:Building-up method : Select edges of the graph one at a

time in such a way that no cycles are created; repeatthis procedure until all vertices are included.

Cutting down method : Choose any cycle and removeany one of its edges; repeat this procedure until nocycles remain.



47

Spanning trees

Problem Use each method to construct a spanning treein the complete graph K 5.



48

Rooted trees

A particular type of a tree structure that appears often isthe rooted tree .



49

Rooted trees: Experiments

Problem Draw the branching tree representing the

outcomes of 2 throws of a six-sided die.

Possible outcomes of experiments can be representedby a branching tree.

Example: tossing a coin.



50

Rooted trees: Games of strategyBranching trees can be used in the analysis of games ,

esp. games of strategy such as chess or tic-tac-toe,and for strategic manoeuvres such as those arising in

military situations.



51

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52

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Approve

the Loan

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Adequate

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Pruned Decision Tree



Revision (and terms to know) Incidence matrices

Types of graphs – null, regular, cycles, Platonic,Petersen, bipartite, path graphs, trees

Complement of a graph Converse of a digraph

Cartesian product (of two graphs) Dual of a graph

Connectivity

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