60 lines
2.8 KiB
Markdown
60 lines
2.8 KiB
Markdown
- #[[MA284 - Discrete Mathematics]]
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- **Previous Topic:** [[Trees]]
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- **Next Topic:**
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- **Relevant Slides:** 
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- In a practical setting, a graph must be stored in some computer-readable format.
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- One of the most common is an **adjacency matrix**.
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- # Adjacency Matrices
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- What is an **adjacency matrix**? #card
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card-last-interval:: -1
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card-repeats:: 1
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card-ease-factor:: 2.5
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card-next-schedule:: 2024-05-30T23:00:00.000Z
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card-last-reviewed:: 2024-05-30T00:17:42.413Z
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card-last-score:: 1
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- If the graph has $n$ vertices, labelled $\{1,2,\cdots, n\}$, then the **adjacency matrix** is an $m \times n$ **binary** matrix, $A$, with entries
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- $$a_{i,j} =
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\begin{cases}
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1 & \text{vertex } i \text{ is adjacent to } j\\
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0 & \text{otherwise.}
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\end{cases}$$
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- ## Properties of the Adjacency Matrix #card
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card-last-interval:: -1
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card-repeats:: 1
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card-ease-factor:: 2.5
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card-next-schedule:: 2024-05-30T23:00:00.000Z
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card-last-reviewed:: 2024-05-30T00:17:59.125Z
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card-last-score:: 1
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- The adjacency matrix of a graph is **symmetric**.
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- If $B = A^k$, then $b_{i,j}$ is the number of paths of length $k$ from vertex $i$ to vertex $j$.
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- We can work out if a graph is connected by looking at the eigenvalues of $A$.
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- If the graphs $G$ & $H$ are isomorphic, and have adjacency matrices $A_G$ & $A_H$, then there is a permutation matrix $P$, such that $PA_GP^{-1}=A_H$.
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- The adjacency matrix idea is easily extended to allow for multigraphs and pseudographs (graphs with loops).
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- For a multigraph, $a_{i,j}$ is the number of edges joining vertices $i$ & $j$.
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- For a pseudograph, $a_{i,i}$ means that there is an edge from the vertex $i$ to itself.
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- # Instance Matrices
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- Graphs can also be represented by an **Incidence Matrix**.
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- If the graph has $v$ vertices, and $e$ edges, then it is an $v \times e$ binary matrix.
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- The rows represent vertices.
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- The columns represent edges.
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- If the matrix is $B = (b_{i,j})$ then $b_{ik} = 1$ means that the vertex $i$ is incident to edge $j$.
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- # Distance Matrices
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- What is the **eccentricity of a vertex**?
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- The **eccentricity of a vertex** is the greatest distance between that vertex & any other vertex in the graph.
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- What is the **radius of a graph**? #card
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card-last-interval:: -1
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card-repeats:: 1
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card-ease-factor:: 2.5
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card-next-schedule:: 2024-05-30T23:00:00.000Z
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card-last-reviewed:: 2024-05-30T00:18:05.726Z
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card-last-score:: 1
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- The **radius of a graph** is the minimum eccentricity of any vertex.
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- What is the **diameter of a graph**?
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- The **diameter of a graph** is the maximum eccentricity of any vertex.
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- This is also the maximum entry in the distance matrix.
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