# 3.7 Recurrence Relations ``` 期中测验时间：
 11月4日
 课件

 ⅠIntroduction to Set Theory
 1.
Sets and Subsets
Representation of set:
 Listing elements, Set builder notion, Recursive
definition
 , , 
 P(A)
 2. Operations on Sets
 Operations and their Properties
 A=?B
 AB, and B A
 Or Properties
 Theorems, examples, and exercises

 3.
Relations and Properties of relations
 reflexive ,irreflexive
 symmetric , asymmetric ,antisymmetric
 Transitive
 Closures of Relations
 r(R),s(R),t(R)=?
 Theorems, examples, and exercises
 4. Operations on Relations
 Inverse relation, Composition
 Theorems, examples, and exercises
5. Equivalence Relation and Partial order relations
 Equivalence Relation
 equivalence class
 Partial order relations and Hasse Diagrams
 Extremal elements of partially ordered sets:
 maximal element, minimal element
 greatest element, least element
 upper bound, lower bound
 least upper bound, greatest lower bound
 Theorems, examples, and exercises

 6.Everywhere
Functions
 one to one, onto, one-to-one correspondence
 Composite functions and Inverse functions
Cardinality, 0.
 Theorems,
examples, and exercises
II
Combinatorics
 1.
Pigeonhole principle
 Pigeon
and pigeonholes
 example，exercise
 2.
Permutations and Combinations
 Permutations
of sets, Combinations of sets
 circular permutation
 Permutations
and Combinations of
multisets
 Formulae
 inclusion-exclusion principle
 generating functions
 integral solutions of the equation
Applications of Inclusion-Exclusion principle
 example,exercise
 Applications generating functions and Exponential
generating functions
 ex=1+x+x2/2!+…+xn/n!+…;
 x+x2/2!+…+xn/n!+…=ex-1;
 e-x=1-x+x2/2!+…+(-1)nxn/n!+…;
 1+x2/2!+…+x2n/(2n)!+…=(ex+e-x)/2;
 x+x3/3!+…+x2n+1/(2n+1)!+…=(ex-e-x)/2;
 examples, and exercises

 3.
recurrence relation
Using Characteristic roots to solve recurrence
relations
 Using Generating functions to solve recurrence
relations
 examples, and exercises

Chapter 5 Graphs
 the
puzzle of the seven bridge in the
K&ouml;nigsberg,
 on the Pregel
 Kirchhoff
 Cayler
CnH2n+1
 The four colour problem四色问题
 Hamiltonian circuits
 1920s,K&ouml;nig: finite and infinite graphs
 OS,Compiler,AI, Network
5.1 Introduction to Graphs
 5.1.1
Graph terminology
Relation: digraph
 Definition 1 : Let V is not empty set. A directed
graph, or digraph, is an ordered pair of sets (V,E)
such that E is a subset of the set of ordered pairs
of V. We denote by G(V,E) the digraph. The
elements of V are called vertices or simply
&quot;points&quot;, and V is called the set of vertices.
Similarly, elements of E are called &quot;edge&quot;, and E
is called the set of edges.

Edge (a,b)
a: initial vertex，
b:terminal vertex
edges (a,b) incident
with the vertices a and
b。
(c,c),(f,f) loop
g: isolated vertex。
 G=(V,E),V={a,b,c,d,e,f,g},
 E={(a,b),(a,c),(b,c),(c,a),(c,c),(c,e),(d,a),(
d,c),(f,e), (f,f)},

Definition 2 ： Let (a,b) be edge in G. The
vertices a and b are called endvertices of edges;
a and b are called adjacent in G; the vertex a is
called initial vertex of edge (a,b), and the vertex
b is called terminal vertex of this edge. The
edge (a,b) is called incident with the vertices a
and b. The edge (a,a) is called loop。The vertex
is called isolated vertex if a vertex is not
g is an isolated vertex, (c,c) ,(f,f) are loop. a

Definition 3: Let V is not empty set. An undirected
graph is an ordered pair of sets (V,E) such that E
is a sub-multiset of the multiset of unordered pairs
of V. We denote by G(V,E) the graph. The
elements of V are called vertices or simply
&quot;points&quot;, and V is called the set of vertices.
Similarly, elements of E are called &quot;edge&quot;, and E
is called the set of edges.
V={v1,v2,v3,v4,v5,v6}，E={{v1,v2},{v1,v5,}，{v2,v2},
{v2,v3},{v2,v4},{v2,v5},{v2,v5},{v3,v4},{v4,v5}}，
edges {v1,v2} incidents with the vertices v1 and v2
loop ; isolated vertex
edge {v2,v5} multiple edge。
4 ： These edges are called
multiple edges if they incident with the
same two vertices. The graph is called
multigraph. The graph is called a simple
graph, if any two vertices in the graph,
may connect at most one edge (i.e., one
edge or no edge) and the graph has no
loop. The complete graph on n vertices,
denoted by Kn, is the simple graph that
contains exactly one edge between each
pair of distinct vertices.
 Definition
 undirected
graph: graph
 finite graph
 finite digraph
 Definition
5：The degree of a vertex
v in an undirected graph is the
number of edges incident with it,
except that a loop at a vertex
contributes twice to the degree of
that vertex. The degree of the vertex
v is denoted by d(v). A vertex is
pendent if only if it has degree one.
The minimum degree of the vertices
of a graph G is denoted by 
(G)(=minvV{d(v)}) and the maximum
degree by  (G)(=maxvV{d(v)}
 b=a,{a,a},
Theorem 5.1(the handshaking theorem): Let G(V,E)
be an undirected graph with e edge.
n
Then:  d (vi )  2e
i 1
 Theorem
5.2: An undirected graph has an
even number of vertices of odd degree.
 Definition
6：In a directed graph the outdegree of a vertex v by d+(v) is the number
of edges with v as their initial vertex. The
in-degree of a vertex v by d-(v), is the
number of edges with v as their terminal
vertex. Note that a loop at a vertex
contributes 1 to both the out-degree and
the in-degree of this vertex. The degree of
the vertex v is denoted by d(v).
 Theorem
5.3: Let G(V,E) be an directed
graph. Then


d
(
v
)

d

 (v) | E |
vV
vV
aD, bB,cA,dE;
(a,b)(D,B), (a,c)(D,A),…，
isomorphism
 Definition
7：The directed graphs G(V,E)
and G'(V',E') be isomorphic if there is a one
to one and onto everywhere function f from
V to V' with the property that (a, b) is an
edge of G if only if (f(a),f(b)) is an edge of G'.
We denote by GG'. The undirected graph
G(V,E) and G'(V',E') be isomorphic if there
is a one to one and onto everywhere function
f from V to V' with the property that {a, b} is
an edge of G if only if {f(a),f(b)} is a edge of
G'. We denote by GG'.
 Petersen
3-regular
The graph is called k-regular if every vertex of
G has degree k.
 Definition
8: Graphs that have a number
assigned to each edge or each vertex
are called weighted graphs
 weighted digraphs
 Definition
9: The graph G'(V',E') is called a
subgraph of G(V,E) If V'V and E'E. If
V'=V, then G'(V',E') is said to be a spanning
subgraph.
Definition 10: If G'(V',E') contains all edges of G
that join two vertices in V' then G' is called the
induced subgraph by V'V and is denoted by
G(V').
 induced subgraph by {v1,v2,v4,v5}




Next: Paths and Circuits,
Connectivity,8.1 P306(Sixth) OR
P291(Fifth)
Exercise P135 27,28; P310 9,10(Sixth);
OR P123 27,28; P295 9,10(Fifth)
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