Discrete Mathematics – Part 4 – Graphs
Nguyễn Thị Thu Vân - May 12, 2023
Relations
Relations
Representing relations
Equivalence relations
Partial orderings
Binary relations. Let 𝐴 and 𝐵 be sets. A
Using Matrices. The relation ℛ can be
A relation on a set A is called an
A relation ℛ on a set 𝑆 is called a partial
binary relation from 𝐴 to 𝐵 is a subset ℛ of
represented by the matrix 𝑀ℛ = [𝑚𝑖𝑗 ],
equivalence relation if it is reflexive,
ordering or partial order if it is reflexive,
𝐴 × 𝐵:
where
symmetric, and transitive.
transitive, and antisymmetric.
ℛ = {𝑎ℛ𝑏 ≡ (𝑎, 𝑏)|𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵}
𝑚𝑖𝑗 = {
1 if (𝑎𝑖 , 𝑏𝑗 ) ∈ ℛ,
Let ℛ be an equivalence relation on a
0 if (𝑎𝑖 , 𝑏𝑗 ) ∉ ℛ
set 𝐴. The set of all elements that are
Members of S are called elements of the
ℛ is symmetric iff 𝑀ℛ = 𝑀ℛ𝑇
related to an
poset. The elements 𝑎 and 𝑏 of a poset
Ex.
element 𝑎 of 𝐴 is called the
(𝑆, ≼) are called comparable if either 𝑎 ≼
exactly one element of 𝐵 is related to each
equivalence class of 𝑎 and is denoted
𝑏 or 𝑏 ≼ 𝑎.
element of 𝐴.
by [𝑎]ℛ .
Example. 𝐴 = {0,1,2}; 𝐵 = {𝑎, 𝑏}. Then ℛ =
{(0, 𝑎), (0, 𝑏), (1, 𝑎), (2, 𝑏)}
o A function represents a relation where
(𝑆, ≼) is a well-ordered set if it is a poset
Let ℛ be an equivalence relation
o A relation on a set 𝐴 is a relation from 𝐴
on a set 𝐴. These statements for
to 𝐴.
Properties of relations.
nonempty subset of 𝑆 has a least element.
elements 𝑎 and 𝑏 of 𝐴 are
𝑎 is maximal in the poset (𝑆, ≼) if there is
equivalent:
no 𝑏 ∈ 𝑆 such that 𝑎 ≺ 𝑏.
(i)
𝑎ℛ𝑏;
a is minimal if there is no element 𝑏 ∈ 𝑆
(ii)
[𝑎] = [𝑏];
such that 𝑏 ≺ 𝑎.
(iii) [𝑎] ∩ [𝑏] = ∅
A relation on a set A is called
such that ≼ is a total ordering and every
Maximal and minimal elements are easy to
spot using a Hasse diagram. They are the
(i) reflexive, i.e. if (𝑎, 𝑎) ∈ 𝑅 for every element 𝑎 ∈ 𝐴.
“top” and “bottom” elements in the
(ii) symmetric, i.e., if (𝑏, 𝑎) ∈ 𝑅 whenever (𝑎, 𝑏) ∈ 𝑅, for all 𝑎, 𝑏 ∈ 𝐴.
diagram.
(iii) transitive, i.e., if whenever (𝑎, 𝑏) ∈ 𝑅 and (𝑏, 𝑐) ∈ 𝑅, then (𝑎, 𝑐) ∈ 𝑅, for all 𝑎, 𝑏, 𝑐 ∈
Ex. The Hasse diagram of the poset
𝐴.
({2,4,5,10,12,20,25}, |). So maximal
(iv) antisymmetric i.e. if (𝑎, 𝑏) ∈ ℛ and (𝑏, 𝑎) ∈ ℛ, then 𝑎 = 𝑏.
elements are 12, 20, and 25 and minimal
𝒏-ary relations. Let 𝐴1 , 𝐴2 , … , 𝐴𝑛 be sets. An n-ary relation on these sets is a subset of 𝐴1 ×
elements are 2 and 5.
𝐴2 × ⋯ × 𝐴𝑛
The sets 𝐴1 , 𝐴2 , … , 𝐴𝑛 are called the domains of the relation, and 𝑛 is called its degree.
Graphs
Graphs
Representing of graph
Type of graphs
Algorithms & Applications
A graph 𝐺 = (𝑉 , 𝐸) consists of 𝑉, a
Adjacency matrix
Null graph - Trivial graph
Breadth first traversal for a graph [BFS]
nonempty set of vertices (or nodes) and 𝐸, a
Adjacency list
Simple graph – Not simple graph
Depth first traversal for a graph [DFS]
set of edges. Each edge has either one or two
Undirected graph – Directed graph
Shortest-path problems [Dijkstra’s shortest
vertices associated with it, called its endpoints.
Complete graph
path algorithm]
An edge is said to connect its endpoints.
Connected / Disconnected graph
Minimum spanning tree
Furthermore, if E consists of of directed edges,
Bipartite graph
Euler paths [Konigberg bridge problems]
(𝑉 , 𝐸) is called directed graph.
Star graph - Weighted graph -
Hamilton paths
Subgraph
Trees
Trees
Properties of trees
A tree is a connected undirected graph with no
A tree with 𝑛 vertices has 𝑛 − 1 edges.
Let 𝐺 be a simple graph. A spanning
Hierarchical structure
simple circuits.
A full 𝑚 −ary tree with
tree of 𝐺 is a subgraph of 𝐺 that is a
Searching efficiency
tree containing every vertex of 𝐺.
Sorting
A rooted tree is a tree in which one vertex has
o 𝑛 vertices has 𝑖 = (𝑛 − 1)/𝑚 internal
been designated as the root and every edge is
vertices and 𝑙 = [(𝑚 − 1)𝑛 + 1]/𝑚
directed away from the root. A rooted tree is
leaves
called an 𝑚-ary tree if every internal vertex
o 𝑖 internal vertices has 𝑛 = 𝑚𝑖 + 1
Spanning trees
Binary search tree
A minimum spanning tree in a
connected weighted graph is a
has no more than 𝑚 children.
vertices and 𝑙 = (𝑚 − 1)𝑖 + 1
spanning tree that has the smallest
The tree is called a full 𝑚-ary tree if every
leaves,
possible sum of weights of its edges.
internal vertex has exactly m children.
o 𝑙 leaves has 𝑛 = (𝑚𝑙 − 1)/(𝑚 − 1)
An 𝑚 −ary tree with 𝑚 = 2 is called a binary
vertices and 𝑖 = (𝑙 − 1)/(𝑚 − 1)
tree.
internal vertices.
Applications of trees