CS1231 Cheat Sheet Part 1

Chapter 3: The Integers Divisibility Write

(

An EULER CIRCUIT is a simple circuit that contains every vertex edge  A connected graph has Euler circuit iff every vertex is of even deg

Binomial

if 



( )

( )

∑( )

( )

( )

( )

( )

( )

∑

( )

∑

An EULER PATH is a simple path which is not a circuit and contains all the edges and vertices  A graph has an Euler path iff it is connected

( ) ( )



Pascal’s Identity:

√ ,

Type Simple Graph Multigraph Pseudograph

Fundamental Theorem of Arithmetic: Every +ve integer > 1 has a divisor which is prime  every +ve integer greater than 1 can be written uniquely as a product of primes. E.g.   An integer ̅ s.t. ̿ The inverse of 

{

}

{

}

{ {

}

(

)

(

)

{

}

Edges

Multiple Edges?

Loops?

Undirected Undirected Undirected

No Yes Yes

No No Yes

An

} {

.

Edges with

as terminal vertex

Edges with COMPLETE BIPARTITE GRAPH denoted

as initial vertex

Chapter 4: Mathematical Induction

Working here to prove that

A tree with vertices has at least two vertices of deg(1) A tree with vertices has edges A full -ary tree ́ internal vertex has vertices Suppose full -ary tree with vertices, internal vertices and leaves, then

-

A rooted -ary tree of height is BALANCED if all leaves are at level or In an -ary tree there are at most vertices at level . If the height is , there are at most leaves.

,

is true

Chapter 5: Counting

With Repetition:

-

are true. Then ….

Thus is true. Therefore, by Mathematical Induction, *proposition* for all *domain*

( )

-ary is FULL if every internal vertex has exactly m children.

An ORDERED ROOTED TREE is a rooted tree in which children of each vertex are ordered. E.g. for T2, the left subtree of is the subtree rooted at while the right subtree if a single vertex

}

is called an inverse of exists iff

Let be the proposition that … Basis Step: is true since… Inductive Step: Assumer that

graph with NO cycles

( )

Chapter 7: Graphs √ is prime

with

and all vertices, but two, have even degrees.

Chapter 8: Trees - A TREE is a connected

Prime Numbers and GCD A positive integer is: - PRIME if it has exactly 2 +ve divisors, 1 and itself; - COMPOSITE if it has more than 2 +ve divisors If is composite, then it has a divisor if does not have a divisor with

No. of paths from

)

-

(

)