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discrete_math [2011/09/22 01:38] – 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. +  * //Adjacent nodes// - nodes that are connected by an arc. 
-  * **Isolated node** - a node not adjacent to any other node. +  * //Loop// - an arc that starts and ends with the same node. 
-  * **Degree** - the number of arcs that end at a node. +  * //Loop free graph// - graph with no loops. 
-  * **Parallel arcs** - two arcs with the same endpoints. +  * //Isolated node// - a node not adjacent to any other node. 
-  * **Simple** - has no parallel arcs and no loops. +  * //Degree// - the number of arcs that end at a node. 
-  * **Complete graph** - all distinct nodes are adjacent (always connected). +  * //Parallel arcs// - two arcs with the same endpoints. 
-  * **Path** - a sequence of consecutive arcs in a graph. +  * //Simple// - has no parallel arcs and no loops. 
-  * **Path length** - the number of arcs in the path. +  * //Complete graph// - all distinct nodes are adjacent (always connected). 
-  * **Connected graph** - there is a path to every node in the graph. +  * //Path// - a sequence of consecutive arcs in a graph. 
-  * **Cycle** - a path exists from a node back to the same node such that no arc is used more than once. +  * //Path length// - the number of arcs in the path. 
-  * **Acyclic** - has no cycles.+  * //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).+//Tree// - a graph that is connected and acyclic (including loops).
  
-  * **Leaves** - nodes with no children. +  * //Leaves// - nodes with no children. 
-  * **Internal nodes** - nodes that are non-leaves. +  * //Internal nodes// - nodes that are non-leaves. 
-  * **Node depth** - the length of the path from the root to the node. +  * //Node depth// - the length of the path from the root to the node. 
-  * **Height** - the greatest depth.+  * //Height// - the greatest depth.
  
-{{:figure2.png?200 |Figure 2}}+<diag> 
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 +</diag>
  
   * Node 1 is the starting node (or root node).   * Node 1 is the starting node (or root node).
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 ---- ----
  
-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.
  
-{{:figure3.png?300|Figure 3}}+<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. 
  
-{{:figure4.png?300|Figure 4}}+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. __//Theorem//__ - a tree with `n` nodes has `(n - 1)` arcs.
Line 76: Line 86:
 {{:figure6.png?100|Figure 6}} {{: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?150 |Figure 7}} {{:figure7.png?150 |Figure 7}}
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 ==== 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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 ==== 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)] 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`
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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.
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-`A * B = C` 
  
 `2xx3 * 3xx4 = 2xx4` `2xx3 * 3xx4 = 2xx4`
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 endamath endamath
  
-Adjacency Matrices (cont.)+==== Adjacency Matrices (cont.) ====
  
 Given the following graph: Given the following graph:
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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.1316655486.txt.gz · Last modified: 2023/08/18 18:15 (external edit)