Chapter 8 graphs

Introduction to Graphs Introduction to Graphs Huynh Tuong Nguyen, Tran Vinh Tan Contents Graph definitions Terminology Special Simple Graphs Representing Graphs and Graph Isomorphism Representing Grap[.]

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Huynh Tuong Nguyen, Tran Vinh Tan

Contents Graph definitions

Terminology

Special Simple Graphs

Representing Graphs and Graph Isomorphism

Discrete Structures for Computing on October 27, 2015

Huynh Tuong Nguyen, Tran Vinh TanFaculty of Computer Science and Engineering

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Terminology

2 Representing Graphs and Graph Isomorphism

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A graph (đồ thị)Gis a pair of(V, E), which are:

• V – nonempty set ofvertices(nodes) (đỉnh)

234

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Undirected Graph (Đồ thị vô hướng)

Definition (Simple graph (đơn đồ thị))

• Each edge connects two different vertices, and

• No two edges connect the same pair of vertices

An edge between two vertices u and v is denoted as {u, v}

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Terminology

Definition (Multigraph (đa đồ thị))

Graphs that may have multiple edges connecting the same vertices

An unordered pair of vertices {u, v} are calledmultiplicity m(bội

m) if it has m different edges between

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Terminology

Definition (Pseudograph (giả đồ thị))

Are multigraphs that have

• loops(khuyên)– edges that connect a vertex to itself

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Terminology

Definition (Directed Graph (đồ thị có hướng))

A directed graphGis a pair of (V, E), in which:

• V – nonempty set of vertices

• E – set of directed edges (cạnh có hướng )

A directed edgestartat u andendat v is denoted as (u, v)

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Terminology

In an undirected graph G = (V, E),

• two verticesuandv ∈ V are calledadjacent(liền kề ) if they

areend-points(điểm đầu mút) of edgee ∈ E, and

• eisincident with(cạnh liên thuộc)uandv

• eis said toconnect (cạnh nối )uandv;

The degree of a vertex

Thedegree of a vertex(bậc của một đỉnh), denoted bydeg(v)is

the number of edges incident with it, except that a loop

contributes twice to the degree of that vertex

• isolatedvertex (đỉnh cô lập): vertex of degree0

• pendantvertex (đỉnh treo): vertex of degree1

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In G, deg(a) = 2, deg(b) = deg(c) = deg(f ) = 4, deg(d) = 1,

Neiborhoods of these vertices are

N (a) = {b, f }, N (b) = {a, c, e, f },

In H, deg(a) = 4, deg(b) = deg(e) = 6, deg(c) = 1,

Neiborhoods of these vertices are

N (a) = {b, d, e}, N (b) = {a, b, c, d, e},

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Theorem (The Handshaking Theorem)

LetG = (V, E) be an undirected graph withmedges Then

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In an directed graph G = (V, E),

• uis said to beadjacent to(nôi tới )v andv is said to be

adjacent from (được nối từ)uif(u, v)is an edge ofG, and

• uis calledinitial vertex(đỉnh đầu) of(u, v)

• v is calledterminal (đỉnh cuối ) orend vertexof(u, v)

• the initial vertex and terminal vertex of a loop are the same

The degree of a vertex

In a graph G with directed edges:

• in-degree(bậc vào) of a vertexv, denoted bydeg−(v), is

the number of edges withv as their terminal vertex

• out-degree(bậc ra) of a vertex v, denoted bydeg+(v), is

the number of edges withv as their initial vertex

Note: a loop at a vertex contributes 1to both the in-degree and

the out-degree of this vertex

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A complete graph (đồ thị đầy đủ ) on n vertices, Kn, is a simple

graph that containsexactly one edge between each pair of distinct

vertices

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A cycle (đồ thị vòng ) Cn, n ≥ 3, consists of n vertices

v1, v2, , vn and edges {v1, v2}, {v2, v3}, , {vn−1, vn}, and

{vn, v1}

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We obtain a wheel (đồ thị hình bánh xe) Wn when we add an

additional vertex to a cycle Cn, for n ≥ 3, and connect this new

vertex to each of the n vertices in Cn

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An n-dimensional hypercube (khối n chiều), Qn, is a graph that

has vertices representing the 2n bit strings of length n Two

vertices are adjacent iff the bit strings that they represent differ in

exactly one bit position

Q1

1110

Q2

101100

111110

Q3

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Applications of Special Graphs

• Local networks topologies

• Star, ring, hybrid

• Parallel processing

• Linear array

• Mesh network

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Terminology

A simple graphGis called bipartite (đồ thị phân đôi ) if its vertex

setV can be partitioned into two disjoint setsV1andV2 such that

every edge in the graph connects a vertex inV1and a vertex inV2

(so that no edge inGconnects either two vertices inV1 or two

vertices inV2)

Example

C6 is bipartite

C6

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Terminology

A complete bipartite Km,nis a graph that

• has its vertex set partitioned intotwo subsetsof m and n

vertices, respectively,

• with an edge between two vertices iff one vertex is in the first

subset and the other is in the second one

K3,3

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Theunion(hợp) of two simple graphsG1= (V1, E1)and

G2= (V2, E2)is a simple graph with vertex setV1∪ V2and edge

setE1∪ E2 The union ofG1andG2is denoted byG1∪ G2

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Adjacency Lists (Danh sách kề)

Vertex Adjacent vertices

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a b c da

bcd

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F

G

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G1= (V1, E1)andG2= (V2, E2)areisomorphic(đẳng cấu) if

there is aone-to-one functionf fromV1 toV2 with the property

that a and b are adjacent inG1iiff (a) andf (b)are adjacent in

G2, for allaandb inV1 Such a function f is called an

isomorphism (một đẳng cấu)

(i.e there is a one-to-one correspondence between vertices of the

two graphs that preserves the adjacency relationship.)

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G1

C

DE

F

G2

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G2

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Determine whether the graphs without loops with the incidence

matrices are isomorphic

• Extend the definition of isomorphism of simple graphs to

undirected graphs containing loops and multiple edges

• Define isomorphism of directed graphs

Tiêu đề	Introduction to Graphs
Tác giả	Huynh Tuong Nguyen, Tran Vinh Tan
Trường học	University of Technology - VNUHCM
Chuyên ngành	Computer Science and Engineering
Thể loại	Chương
Năm xuất bản	2015
Thành phố	Ho Chi Minh City

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