Introduction to networks & matrices
What is a network?
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A network is a weighted graph
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A network is used to model real-world situations
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The weight of an arc usually represents a measure such as distance or time
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What is a matrix?
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A matrix can be used to represent a graph or network in the form of a table
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The elements within the matrix give information about the connections between the different vertices
Networks & matrices
What is an adjacency matrix?
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An adjacency matrix is a square matrix where all of the vertices in the graph are listed as the headings for both the rows and columns
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An adjacency matrix can be used to show the number of direct connections between two vertices
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An entry of 0 in the matrix means that there is no direct connection between that pair of vertices
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In a simple graph the only entries are either 0 or 1
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A loop is indicated in an adjacency matrix with a value in the leading diagonal (the line from top left to bottom right)
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In an undirected matrix the value in the leading diagonal will be 2 because you can use the loop to travel out of and into the vertex in two different directions
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In a directed matrix, if the loop has been given a direction, the value in the leading diagonal will be 1 as you can only travel along the loop out of and back into the vertex in one direction
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For a graph with no loops every entry in the leading diagonal will be 0
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An undirected graph will be symmetrical in the leading diagonal
What is a distance matrix?
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A distance matrix is different to an adjacency matrix as the value in each cell is the weight of the edge connecting that pair of vertices
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Weight could be cost, distance, time etc.
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An empty cell can be used to indicate that there is no connection between a pair of vertices
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A directed network is not symmetrical along the leading diagonal (the line from top left to bottom right)
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When drawing a network from its distance matrix be careful when labelling the edges
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For an un-directed graph the two cells between a specific pair of vertices will be the same so connect the vertices with one edge labelled with the relevant weight
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For a directed graph if the two cells between a specific pair of vertices have different values draw two lines between the vertices and label each with the correct weight and direction
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A distance matrix can be used to work out the weight of different walks in the network
Worked Example
a) Write down the adjacency matrix for the graph shown below.
<img alt=”table row blank cell table row bold A bold B bold C bold D bold E end table end cell row cell table row bold A row bold B row bold C row bold D row bold E end table end cell cell stretchy left parenthesis table row 0 1 1 0 1 row 1 0 0 0 0 row 1 0 1 1 1 row 0 0 0 0 1 row 2 1 1 1 2 end table stretchy right parenthesis end cell end table” data-mathml='<math ><semantics><mtable><mtr><mtd><mrow></mrow></mtd><mtd><mtable><mtr><mtd><mi mathvariant=”bold”>A</mi></mtd><mtd><mi mathvariant=”bold”>B</mi></mtd><mtd><mi mathvariant=”bold”>C</mi></mtd><mtd><mi mathvariant=”bold”>D</mi></mtd><mtd><mi mathvariant=”bold”>E</mi></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mi mathvariant=”bold”>A</mi></mtd></mtr><mtr><mtd><mi mathvariant=”bold”>B</mi></mtd></mtr><mtr><mtd><mi mathvariant=”bold”>C</mi></mtd></mtr><mtr><mtd><mi mathvariant=”bold”>D</mi></mtd></mtr><mtr><mtd><mi mathvariant=”bold”>E</mi></mtd></mtr></mtable></mtd><mtd><mstyle mathvariant=”bold”><mo stretchy=”true”>(</mo><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable><mo stretchy=”true”>)</mo></mstyle></mtd></mtr></mtable><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″,”autoformat”:true}</annotation></semantics></math>’ data-type=”finalAnswer” height=”170″ role=”math” 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