MATH

Question 1

A collection of points called vertices or
nodes and line segments or curves called edges
that connect the vertices

Answer 1

GRAPH

Question 1

is an edge connecting a vertex to itself.
a special type of edge.
● It can only have one vertex.

Answer 1

LOOP

Question 2

A complete graph is a graph that has an edge
connecting every pair of vertices.

Answer 2

COMPLETE GRAPH

Question 3

Two vertices are adjacent if there is an edge joining
them

Answer 3

ADJACENT (VERTICES)

Question 4

Graphs are considered ________if they have the
same vertices connected in the same way

Answer 4

equivalent

Question 5

an alternating sequence of vertices and
edges. It can be seen as a trip from one vertex to
another using the edges of a graph.

● If a path begins and ends with the same vertex,
it is a closed path or a circuit or cycle.

Answer 5

PATH

Question 6

A graph is connected if for any two ________, there
is at least one path that joins them. A _________ is an
edge that when you remove makes the graph
disconnected.

Answer 6

Vertices

Bridge

Question 7

The degree of a vertex is the number of edges
attached to it.

Answer 7

DEGREE.

Question 8

The idea is to color the vertices or
edges of a graph in such a way that adjacent
vertices or concurrent edges are given different
colors.

Answer 8

GRAPH COLORING

Question 8

The smallest number of colors required to color the
vertices of a graph.

Answer 8

CHROMATIC NUMBER OF THE GRAPH

Question 8

FINDING THE TOTAL NUMBER OF EDGES IN A
COMPLETE GRAPH

Answer 8

𝑒𝑛 =1/2 x 𝑛(𝑛 − 1)

Question 9

A graph is 2-colorable if and only if it has _________
that consist of an ____number of vertices

Answer 9

No circuits

Odd

Question 10

A map can be represented by a graph with the
different regions as vertices. Two regions are
adjacent if they share part of their boundaries.

Answer 10

MAP COLORING

Question 11

● A path that passes through every edge exactly
once only
● A path that contains all the edges of the graph
○ A graph has an Euler path if and only if no
more than two of its vertices have odd degree

Answer 11

EULER PATH

Question 12

a graph is a closed path that
contains all the edges of the graph. A graph that
has an Euler circuit is called Eulerian.

Answer 12

EULER CIRCUIT

Question 13

If all vertices are even, the graph has at least
__________

If exactly two vertices are odd, the graph has
_________

If there are more than two odd vertices, the
graph has __________

Answer 13

1.) one Euler circuit
2.) no Euler circuits but at least one Euler path
3.) no Euler path and no Euler circuits

Question 14

a graph is a path passing through each vertex of the graph exactly once.

● If the path is closed, it is called a Hamiltonian cycle.

● If a graph has a Hamiltonian cycle, it is called
Hamiltonian.

Answer 14

HAMILTONIAN PATHS AND CYCLES

Question 15

Every complete graph with more than two vertices
has a ________

Answer 15

Hamiltonian circuit

Question 16

Two Hamiltonian circuits are the _____if they
pass through the same vertices in the same order,
regardless of the vertex where they begin and end

Answer 16

Same

Question 17

a graph in which any two vertices are
connected by exactly one path.

Are graphs that do not contain even a single cycle.
They represent hierarchical structure in a graphical
form

Answer 17

TREES

Question 18

1. A tree has no circuits.
2. Trees are connected graphs.
3. Every edge in a tree is a bridge.
4. A tree with n vertices has exactly n-1 edges.

Answer 18

PROPERTIES OF TREES

Question 19

an undirected, disconnected, acyclic
graph. In other words, a disjoint collection of trees
is known as _____. Each component of a _____is a
tree.

Answer 19

FOREST

Question 20

a tree that results
from the removal of as many edges as possible
from the original graph without making it
disconnected.

Answer 20

SPANNING TREE

Question 21

the problem of dividing up a fixed
number of things or objects among groups of
different sizes.

a method of dividing a
population into several parts

Answer 21

Apportionment

Question 22

1.) the ratio between the total population and the
total number to apportion

2.) the whole number part of the quotient of a
population divided by the standard divisor

3.) the standard quota rounded down to a whole
number

4.)
○ the standard quota rounded up to a whole
number.
● +1 on the whole number

5.)
○ the standard quota rounded off to a whole
number
● how we usually round off numbers. if the
number after the decimal point is ,<5, it stays
the same; if it’s >5, round up.

Answer 22

● Standard Divisor
● Standard Quota
● Standard Quota Lower (Lower Quota)
● Standard Quota Upper (Upper Quota)
● Standard Quota Round Off (Round Off Quota)

Question 23

The first census was to be taken in ___

Answer 23

1790

Question 24

approved by the U.S. Congress in 1791, but
was vetoed by President Washington.

Answer 24

HAMILTON’S METHOD

Question 25

Proposed a new apportionment
method after President Washington vetoed
Hamilton’s Method in 1791.

● This method tends to favor large states.

Answer 25

Thomas Jefferson

Question 26

Proposed by former U.S. President John Quincy
Adams

Quota is rounded up to the nearest whole number
instead of following normal rounding rules

Answer 26

ADAMS’ METHOD

Question 27

It was first proposed by Daniel Webster
(1782-1852)

This method rounds the quota to the nearest
whole number instead of dropping it.

The Modified Standard Divisor (MSD) must be
less than the Standard Divisor (SD)

If the total of the rounded sub-quotas meets or
exceeds the required number, the MSD must be
greater than the SD.

Answer 27

WEBSTER’S METHOD

Question 28

credited to Edward Vermilye Huntington and Joseph Adna Hill.

Rounding depends on comparing the quota with its
geometric mean.

Answer 28

HUNTINGTON-HILL METHOD

Question 29

A method for a group, such as a meeting
or an electorate, in order to make a collective
decision or express an opinion usually following
discussions, debates or election campaigns.

Answer 29

VOTING

Question 30

• is a ballot which the voter ranks
  the choices in order of preference.

Answer 30

Preference ballot

Question 30

• a table that provides
  information about the number and how voters
  would rank the alternatives if their first choice was
  unsuccessful.

Answer 30

Preference schedule

Question 31

● Each person votes for a candidate.
● Candidates who get the majority vote wins.
● If no candidate gets the majority vote, the candidate
with the fewest votes is eliminated and a new
election is held.
● Process of elimination.

Answer 31

PLURALITY WITH ELIMINATION WITHOUT RANK

Question 32

● Each voter votes for one candidate.
● The candidate with the most votes wins.
● The winning candidate does not have to have the
majority vote (More than 50% of the total vote).

Answer 32

PLURALITY METHOD OF VOTING

Question 33

● Also called the Instant Runoff Method.
● First, Eliminate the candidate with the fewest
number of first-place voters.
● If two or more of the alternatives have the same
number of first-place votes, all are eliminated unless
that would eliminate all alternatives

Answer 33

PLURALITY WITH ELIMINATION WITH RANK

Question 34

An isomorphism between two graphs means that
they have essentially the same structure.

Answer 34

ISOMORPHISM

Question 35

if you can combine two edges by deleting a vertex to
combine two edges (or smoothing out the edges)
and repeating the process until you make two
graphs the same

Answer 35

HOMOMORPHISM

Question 36

An algebraic system is a mathematical system
consisting of a set of elements called the domain
and one or more operations for combining the
elements on the domain

Answer 36

GROUP THEORY

Question 37

Refers to the degree with which an entity
(object, function, etc) is invariant under some
geometric operation (such as rotation about an
axis, reflection through a plane, and inversion about
a point.)

Answer 37

SYMMETRY GROUPS