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# FM Graphs and Networks

## 1.3 Introduction to Network Matrices

### Network Matrices

• Matrices provide an alternative means to model networks. Using matrices also allows for more systemic methods for numerical analysis of networks.
• A matrix is similar in form to a table, with a number of elements arranged into rows and columns. The vertices are listed along the rows and columns of the matrix. Each element is representative of the connection between the vertices that it’s corresponding row and column represent.
• Each element indicates the number of edges which directly connect the vertex corresponding to the row to the vertex corresponding to the column.
• If the graph is not a digraph, the matrix will be symmetric.

Note: network matrices are often known as adjacency matrices

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## 1.2 Further Concepts for Network Graphs

Note: if you cannot remember the basics of network graphs, revise notes for 1.1 Basics of Network Graphs.

### Simple Graph

• Graphs which do not contain loops or duplicate edges (i.e. there are no two vertices connected directly by more than one edge).
• Each vertex has a maximum degree of n-1.
• A simple graph has a maximum number of edges of \frac {n(n-1)}{2}.
• Graphs which contain loops or duplicate edges are sometimes called multigraphs.

Example

### Complete Graph

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## 1.1 Basics of Network Graphs

### Network Graphs

• A network graph shows the connections between multiple individuals or locations.
• It consists of a number of dots, known as vertices (singular: vertex), each representing an individual or location, connected by lines, known as edges.
• If two vertices are connected by an edge, this indicates the two individuals or locations the vertices represent are directly connected.
• Examples of situations which can be modelled by network graphs include communication networks (showing which individuals can communicate with each other) and transportation networks (showing which towns are connected via roads, train tracks, etc).

Note: vertices are sometimes known as nodes.

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