====== Discrete Math ======

===== Graphs =====

`G = (:N, A, f:)`

  * `N` - non-empty set of nodes
  * `A` - set of arcs
  * `f` - function that associates each arc with an unordered pair of endpoints

`G = (:{1,2,3,4,5,6}, {a,b,c,d,e}, {(a, 2-3), (b, 1-4), (c, 4-5), (d, 1-3), (e, 2-2)}:)`

<diag>
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</diag>

  * //Adjacent nodes// - nodes that are connected by an arc.
  * //Loop// - an arc that starts and ends with the same node.
  * //Loop free graph// - graph with no loops.
  * //Isolated node// - a node not adjacent to any other node.
  * //Degree// - the number of arcs that end at a node.
  * //Parallel arcs// - two arcs with the same endpoints.
  * //Simple// - has no parallel arcs and no loops.
  * //Complete graph// - all distinct nodes are adjacent (always connected).
  * //Path// - a sequence of consecutive arcs in a graph.
  * //Path length// - the number of arcs in the path.
  * //Connected graph// - there is a path to every node in the graph.
  * //Cycle// - a path exists from a node back to the same node such that no arc is used more than once.
  * //Acyclic// - has no cycles.

----

//Tree// - a graph that is connected and acyclic (including loops).

  * //Leaves// - nodes with no children.
  * //Internal nodes// - nodes that are non-leaves.
  * //Node depth// - the length of the path from the root to the node.
  * //Height// - the greatest depth.

<diag>
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</diag>

  * Node 1 is the starting node (or root node).
  * Node 1 is the parent node.
  * Nodes 2 and 3 are children of 1.
  * Nodes 4, 5, and 6 are children of 2.
  * Nodes 7 and 8 are children of 3.
  * a and b are branches.

----

A //binary tree// is a tree where each node has at most two children.

<diag>
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</diag>


A //full binary tree// is a binary tree where all internal nodes have two children and all leaves of the tree are at the same depth.

<diag>
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</diag>

__//Theorem//__ - a tree with `n` nodes has `(n - 1)` arcs.
==== Undirected Graphs ====

`G = (:N, A, f :)`

`G = (:{a, b, c}, {1, 2}, {(1, a-c), (2, c-b)}:)`

This graph is described as: the set of nodes, `a`, `b` and `c`; the set of arcs, `1` and `2`; arc `1` spans nodes `a` and `c`, arc `2` spans nodes `c` and `b`.

{{:figure5.png?100|Figure 5}}

==== Directed Graphs ====

Directed graphs are identified by a function that associates an arc with an __ordered pair__ of nodes.

`G = (:{a, b, c}, {1, 2}, {(1, (a, c)), (2, (c, b))}:)`

This graph is described as: the set of nodes, `a`, `b` and `c`; the set of arcs, `1` and `2`; arc `1` is directed from node `a` to node `c`, arc `2` is directed from node `c` to node `b`.

{{:figure6.png?100|Figure 6}}

  * //Initial node// - the node from which an arc begins.
  * //Terminal node// - the node where an arc ends.
  * //Adjacent node// - in a directed graph, node `b` is adjacent to node `a` if there is an arc from node `a` to node `b`.
  * //Inverted arc// - if there is an arc from node `a` to node `b`, an inverted arc would produce the opposite association from node `b` to node `a`.

{{:figure7.png?150 |Figure 7}}

Attributes of this graph:
  * Nodes `b` and `c` are adjacent to node `a`. Node `a` is not adjacent to nodes `b` and `c`.
  * Node `a` is adjacent only to node `d`.
  * Arc `5` is the only loop.
  * Arcs `2`, `3` and `4` form a cycle.
  * The degree of node `a` is 3; 2 arcs out, 1 arc in.

\\
\\
\\

----

Parallel arcs - arcs that have the same initial node and same terminal node.

{{:figure8.png?150|Figure 8}}

Complete graph - all distinct nodes are adjacent.

{{:figure9.png?300|Figure 9}}

Connected graph - a path exists from any node to any other node.

{{:figure10.png?200|Figure 10}}

==== Adjacency Matrices ====

//Adjacency matrix//  - the number of arcs from one node to another node.

An adjacency matrix describes which nodes in a graph are adjacent to which other nodes.

`G = (:{a, b, c}, {1, 2}, {(1, a-b), (2, b-c)}:)`

{{:figure11.png?150|Figure 11}}
{{:figure12.png?200|Figure 12}}

Matrix dimensions are the number of rows by the number of columns.

E.g. 2 rows X 3 columns: `[(1, 0, 1),(0, 1, 1)]`

An adjacency matrix always has the same number of rows as columns.

{{:figure13.png?200|Figure 13}}

The __main diagonal__ runs from the upper left to lower right corners of the matrix. The main diagonal will always have 0's in each position unless the graph has a loop. A loop in the graph will be indicated by a 1 on the main diagonal.

For a __symmetric matrix__, each side of the main diagonal is a mirror image of the other.

An undirected graph always has a symmetric adjacency matrix.

==== Matrix Multiplication ====

  * //Square Matrix// - a matrix with the same number of rows and columns.
  * //Zero Matrix// - a matrix with all zeros.

`[(0, 0, 0),(0, 0, 0)]`

//Identity Matrix//  - a matrix with all 1'a along the main diagonal with 0's in all other positions. An identity matrix must be square.

Identity matrix: `I = [(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)] 4 xx 4`

Not identity matrix: `[(1, 0, 0, 0), (0, 1, 0, 0)] 2 xx 4`

`I * A = A`

Matrix multiplication is NOT commutative. Order does make a difference.

`A * B != B * A`
=== Scalar multiplication of matrices ===

`2 * [(a_(11), a_(12)), (a_(21), a_(22))] = [(2 * a_(11), 2 * a_(12)), (2 * a_(21), 2 * a_(22))]`
=== Product of matrices ===

`A * B = C`

The number of columns in `A` must be the same as the number of rows in `B` to multiply.

`2xx3 * 3xx4 = 2xx4`

amath
[(a_(11), a_(12)), (a_(21), a_(22))] *
[(b_(11), b_(12)), (b_(21), b_(22))] =
[(a_(11) * b_(11) + a_(12) * b_(21), a_(11) * b_(12) + a_(12) * b_(22)), (a_(21) * b_(11) + a_(22) * b_(21), a_(21) * b_(12) + a_(22) * b_(22))] =
[(c_(11), c_(12)), (c_(21), c_(22))]
endamath

==== Adjacency Matrices (cont.) ====

Given the following graph:

`G = (:{a, b, c}, {1, 2}, {(1, a-b), (2, b-c)}:)`

{{:figure11.png?150|Figure 14a}}
{{:figure14.png?200|Figure 14b}}

`A * A = A^2 = [(0, 1, 0), (1, 0, 1), (0, 1, 0)] * [(0, 1, 0), (1, 0, 1), (0, 1, 0)] =
[(0*0+1*1+0*0, 0*1+1*0+0*1, 0*0+1*1+0*0),(1*0+0*1+1*0, 1*1+0*0+1*1, 1*0+0*1+1*0), (0*0+1*1+0*0, 0*1+1*0+0*1, 0*0+1*1+0*0)] =
[(1, 0, 1), (0, 2, 0), (1, 0, 1)]`

{{:figure15.png?200|Figure 15}}

`A` represents the number of paths of length 1.
  * There is one path from node `a` to node `b`.
  * Likewise, there is one path from node `b` to node `a`.
  * There is one path from node `b` to node `c`.
  * There is one path from node `c` to node `b`.

`A^2` represents the number of paths of length 2.
There is one path of length 2 from node `a` to node `a`. The path traverses from node `a` to node `b` then back to node `a`.
  * There is one path of length 2 from node `c` to node `c`.
  * There are two paths of length 2 from node `b` to node `b`. The first traverses from node `b` to node `a` then back to node `b`. The second traverses from node `b` to node `c` then back to node `b`.
  * There is one path of length 2 from node `a` to node `c`. The path traverses from node `a` to node `b` then continues on to node `c`.
  * There is one path of length 2 from node `c` to node `a`.

`A^n` represents the number of paths of length `n` between any two nodes.

In determining the value of `n`, consider the number of distinct nodes - 1 (not including isolated nodes).

`R = A + A^2 + A^3 + ... + A^n`

`R` - The number of paths `\le` length `n`