Graphs Lec1

8/13/2019 Graphs Lec1

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GRAPHSLecture 7


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Graph: is a pair of sets (V,E), where

elements ofEconnects elements of V.

Elements of Vare called as vertices/nodes

-the number of vertices is called the order

of G

The elements ofEare the edges, or arcs

-the number of edges is called thesize of

G

The edge e= (v1, v

2) has as endpointsv

1

and v2. eis incidentfromv1and incident

on v2. v1and v2are adjacentor neighbors.

v1is the initialvertex and v2is the

terminalvertex.


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Graph types

Multigraph: Graph with multiple edges

between the same pair of vertices.

Simple graph: No multiple edges and no

loops;Eis a subset of VV.

Directed/ undirectedgraph: Edges go in

a specific direction, or are always

symmetric. Combinations of the above: for example,

directed simple graph


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Degree of vertices

The degreeof a vertex v,deg(v) is the number ofedges with vas an endpoint, except that a loopadds 2 to the degree.

If the graph is directed, thein-degreeof v

deg+

(v) is the number of edges with vasterminal vertex. Similarly, the out-degreeof vdeg-(v) is the number of edges with vas initialvertex.

If the degree of all the vertices of the graph isthe same, k , then the graph is called as k-regular

graph. e.g 3-regular graph is calleda cubicgraph.


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Degree

If di=deg vi; d1,d2,dnis the degree

sequenceof G.

Convention: Label the vertices such that

is monotone increasing sequence

(G) = d1 d2 dn=(G)


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Handshaking Theorem

If V={v1,v2..vn} is a vertex set of a non

directed graph G, then

For directed graph


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Prove that for any non directed graph

there is an even number of vertices with

odd degree using the handshaking

theorem.


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Path Types

In a non directed graph G a sequence of zero or more edgesof the form {vo,v1}{v1,v2}{vn-1,vn} is called asPathfrom voto vn

Vertex vois called as initial vertex.

Vertex vnis called the terminal vertex.

If vo= vnpath is called closed path and if vo vnpath is calledopen path.

A path is said to be trivialif there is no edges at all in thepath, such a path is consisting of a singleton set {vo}

Simple path:-No repetition of edges and vertices.

Circuit:- No repetition of edges and whose endpoints are thesame.

Cycle:-No repetition of edges and vertices and whoseendpoints are the same.


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Show that the sum, over the set of people atthe party, of the number of people a personhas shaken hands with, is even. Assume thatno one shakes his or her own hand.

In a party of four couples, severalhandshakes took place. No one shook handswith himself neither with his own spouse

and no one shook hands with the sameperson more than once .

After all the handshakes have completedhow many hands each person has shaken?


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Does there exist a simple graph withfive vertices of these degrees? If so,

draw such a graph.

3,3,3,3,2 1,2,3,4,5

1,2,3,4,4

3,4,3,4,3 0,1,2,2,3

1,1,1,1,1


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Subgraph

A graph G = (V, E) is a subgraph ofG = (V, E) if V V and E E (V

V)

G is a proper subgraph if G G.

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