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Discrete MathematicsLecture 9

Alexander Bukharovich New York University

Graphs

- Graph consists of two sets set V of vertices and

set E of edges. - Terminology endpoints of the edge, loop edges,

parallel edges, adjacent vertices, isolated

vertex, subgraph, bridge edge - Directed graph (digraph) has each edge as an

ordered pair of vertices

Special Graphs

- Simple graph is a graph without loop or parallel

edges - A complete graph of n vertices Kn is a simple

graph which has an edge between each pair of

vertices - A complete bipartite graph of (n, m) vertices

Kn,m is a simple graph consisting of vertices,

v1, v2, , vm and w1, w2, , wn with the

following properties - There is an edge from each vertex vi to each

vertex wj - There is no edge from any vertex vi to any vertex

vj - There is no edge from any vertex wi to any vertex

wj

The Concept of Degree

- The degree of a vertex deg(v) is a number of

edges that have vertex v as an endpoint. Loop

edge gives vertex a degree of 2 - In any graph the sum of degrees of all vertices

equals twice the number of edges - The total degree of a graph is even
- In any graph there are even number of vertices of

odd degree

Exercises

- Two jugs have capacities of of 3 and 5 gallons.

Can you use these jugs to measure out exactly one

gallon? - Bipartite graphs
- Complement of a graph
- What is the relationship between the number of

edges between a graph and its complement - Can there be a simple graph that has vertices

each of different degree? - In a group of two or more people, must there be

at least two people who are acquainted with the

same number of people?

Paths and Circuits

- A walk in a graph is an alternating sequence of

adjacent vertices and edges - A path is a walk that does not contain a repeated

edge - Simple path is a path that does not contain a

repeated vertex - A closed walk is a walk that starts and ends at

the same vertex - A circuit is a closed walk that does not contain

a repeated edge - A simple circuit is a circuit which does not have

a repeated vertex except for the first and last

Connectedness

- Two vertices of a graph are connected when there

is a walk between two of them. - The graph is called connected when any pair of

its vertices is connected - If graph is connected, then any two vertices can

be connected by a simple path - If two vertices are part of a circuit and one

edge is removed from the circuit then there still

exists a path between these two vertices - Graph H is called a connected component of graph

G when H is a subgraph of G, H is connected and H

is not a subgraph of any bigger connected graph - Any graph is a union of connected components

Euler Circuit

- Euler circuit is a circuit that contains every

vertex and every edge of a graph. Every edge is

traversed exactly once. - If a graph has Euler circuit then every vertex

has even degree. If some vertex of a graph has

odd degree then the graph does not have an Euler

circuit - If every vertex of a graph has even degree and

the graph is connected then the graph has an

Euler circuit - A Euler path is a path between two vertices that

contains all vertices and traverces all edge

exactly ones - There is an Euler path between two vertices v and

w iff vertices v and w have odd degrees and all

other vertices have even degrees

Hamiltonian Circuit

- Hamiltonian circuit is a simple circuit that

contains all vertices of the graph (and each

exactly once) - Traveling salesperson problem

Exercises

- For what values of m and n, does the complete

bipartite graph of (m, n) vertices have an Euler

circuit, a Hamiltonian circuit? - What is the maximum number of edges a simple

disconnected graph with n vertices can have? - Show that a graph is bipartite iff it does not

have a circuit with an odd number of edges

Matrix Representation of a Graph

- Adjacency matrix
- Undirected graphs and symmetric matrices
- Number of walks of a particular length between

two vertices

Isomorphism of Graphs

- Two graphs G (V, E) and G (V, E) are

called isomorphic when there exist two bijective

functions g V ? V and h E ? E so that if v

is an endpoint of e iff g(v) is an endpoint of

h(e) - Property P is called an isomorphic invariant when

given any two isomorphic graphs G and G, G has

property P, then G has property P as well - The following properties are isomorphic

invariants - Number of vertices, number of edges
- Number of vertices of a particular degree
- Connectedness
- Possession of a circuit of a particular length
- Possession of Euler circuit, Hamiltonian

circuit22222

Trees

- Connected graph without circuits is called a tree
- Graph is called a forest when it does not have

circuits - A vertex of degree 1 is called a terminal vertex

or a leaf, the other vertices are called internal

nodes - Decision tree
- Syntactic derivation tree
- Any tree with more than one vertex has at least

one vertex of degree 1 - Any tree with n vertices has n 1 edges
- If a connected graph with n vertices has n 1

edges, then it is a tree

Rooted Trees

- Rooted tree is a tree in which one vertex is

distinguished and called a root - Level of a vertex is the number of edges between

the vertex and the root - The height of a rooted tree is the maximum level

of any vertex - Children, siblings and parent vertices in a

rooted tree - Ancestor, descendant relationship between vertices

Binary Trees

- Binary tree is a rooted tree where each internal

vertex has at most two children left and right.

Left and right subtrees - Full binary tree
- Representation of algebraic expressions
- If T is a full binary tree with k internal

vertices then T has a total of 2k 1 vertices

and k 1 of them are leaves - Any binary tree with t leaves and height h

satisfies the following inequality t ? 2h

Spanning Trees

- A subgraph T of a graph G is called a spanning

tree when T is a tree and contains all vertices

of G - Every connected graph has a spanning tree
- Any two spanning trees have the same number of

edges - A weighted graph is a graph in which each edge

has an associated real number weight - A minimal spanning tree (MST) is a spanning tree

with the least total weight of its edges

Finding Minimal Spanning Tree

- Kruskals algorithm
- Prims algorithm

Exercises

- If all edges in a graph have different weights,

does this graph have a unique MST?

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