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discrete_math [2011/09/22 00:37] – javapimpdiscrete_math [2023/08/18 18:15] (current) – external edit 127.0.0.1
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 `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)}:)` `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)}:)`
  
-{{:figure1.png?200|Figure 1}}+<diag> 
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-  * Adjacent nodes - nodes that are connected by an arc. +</diag>
-  * 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).+
  
-{{:figure2.png?200|Figure 2}}+  * //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 starting node (or root node).
Line 36: Line 49:
   * Nodes 7 and 8 are children of 3.   * Nodes 7 and 8 are children of 3.
   * a and b are branches.   * a and b are branches.
-{{:figure2.png?300 |}} 
-  * 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. 
  
-A __binary tree__ is a tree where each node has at most two children.+---- 
 + 
 +A //binary tree// is a tree where each node has at most two children. 
 + 
 +<diag> 
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 +</diag>
  
-{{:figure3.png?200|}} 
  
-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.+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.
  
-{{:figure4.png?200|}}+<diag> 
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 +</diag>
  
-''__Theorem__ - a tree with `n` nodes has `(n - 1)` arcs.''+__//Theorem//__ - a tree with `n` nodes has `(n - 1)` arcs.
 ==== Undirected Graphs ==== ==== Undirected Graphs ====
  
Line 59: Line 74:
 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`. 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?200|}}+{{:figure5.png?100|Figure 5}}
  
 ==== Directed Graphs ==== ==== Directed Graphs ====
Line 69: Line 84:
 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`. 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?200|}}+{{:figure6.png?100|Figure 6}}
  
-  * Initial node - the node from which an arc begins. +  * //Initial node// - the node from which an arc begins. 
-  * Terminal node - the node where an arc ends. +  * //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`. +  * //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`.+  * //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?200|}}+{{:figure7.png?150 |Figure 7}}
  
 Attributes of this graph: Attributes of this graph:
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   * Arcs `2`, `3` and `4` form a cycle.   * Arcs `2`, `3` and `4` form a cycle.
   * The degree of node `a` is 3; 2 arcs out, 1 arc in.   * 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. Parallel arcs - arcs that have the same initial node and same terminal node.
  
-{{:figure8.png?200|}}+{{:figure8.png?150|Figure 8}}
  
 Complete graph - all distinct nodes are adjacent. Complete graph - all distinct nodes are adjacent.
  
-{{:figure9.png?200|}}+{{:figure9.png?300|Figure 9}}
  
 Connected graph - a path exists from any node to any other node. Connected graph - a path exists from any node to any other node.
  
-{{:figure10.png?200|}}+{{:figure10.png?200|Figure 10}}
  
 ==== Adjacency Matrices ==== ==== Adjacency Matrices ====
  
-Adjacency matrix - the number of arcs from one node to another node.+//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. An adjacency matrix describes which nodes in a graph are adjacent to which other nodes.
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 `G = (:{a, b, c}, {1, 2}, {(1, a-b), (2, b-c)}:)` `G = (:{a, b, c}, {1, 2}, {(1, a-b), (2, b-c)}:)`
  
-{{:figure11.png?200|}} +{{:figure11.png?150|Figure 11}} 
-{{:figure12.png?200|}}+{{:figure12.png?200|Figure 12}}
  
 Matrix dimensions are the number of rows by the number of columns. Matrix dimensions are the number of rows by the number of columns.
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 An adjacency matrix always has the same number of rows as columns. An adjacency matrix always has the same number of rows as columns.
  
-{{:figure13.png?200|}}+{{: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. 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.
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 ==== Matrix Multiplication ==== ==== Matrix Multiplication ====
  
-  * Square Matrix - a matrix with the same number of rows and columns. +  * //Square Matrix// - a matrix with the same number of rows and columns. 
-  * Zero Matrix - a matrix with all zeros.+  * //Zero Matrix// - a matrix with all zeros.
  
 `[(0, 0, 0),(0, 0, 0)]` `[(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//  - 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)]&nbsp;4 xx 4`+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)]&nbsp;2 xx 4`+Not identity matrix: `[(1, 0, 0, 0), (0, 1, 0, 0)] 2 xx 4`
  
 `I * A = A` `I * A = A`
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 The number of columns in `A` must be the same as the number of rows in `B` to multiply. The number of columns in `A` must be the same as the number of rows in `B` to multiply.
  
-`&nbsp;A&nbsp;&nbsp;*&nbsp;&nbsp;B&nbsp;&nbsp;=&nbsp;&nbsp;C` +`2xx3 * 3xx4 = 2xx4`
- +
-`2xx3&nbsp;&nbsp;3xx4&nbsp;&nbsp;&nbsp;2xx4`+
  
 amath amath
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 endamath endamath
  
-Adjacency Matrices (cont.)+==== Adjacency Matrices (cont.) ====
  
 Given the following graph: Given the following graph:
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 `G = (:{a, b, c}, {1, 2}, {(1, a-b), (2, b-c)}:)` `G = (:{a, b, c}, {1, 2}, {(1, a-b), (2, b-c)}:)`
  
-{{:figure11.png?200|}} +{{:figure11.png?150|Figure 14a}} 
-{{:figure14.png?200|}}+{{: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)] = `A * A = A^2 = [(0, 1, 0), (1, 0, 1), (0, 1, 0)] * [(0, 1, 0), (1, 0, 1), (0, 1, 0)] =
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 [(1, 0, 1), (0, 2, 0), (1, 0, 1)]` [(1, 0, 1), (0, 2, 0), (1, 0, 1)]`
  
-{{:figure15.png?200|}}+{{:figure15.png?200|Figure 15}}
  
 `A` represents the number of paths of length 1. `A` represents the number of paths of length 1.
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 `R = A + A^2 + A^3 + ... + A^n` `R = A + A^2 + A^3 + ... + A^n`
  
-`R` - The number of paths `<=` length `n`+`R` - The number of paths `\le` length `n`
  
  
discrete_math.1316651825.txt.gz · Last modified: 2023/08/18 18:15 (external edit)