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Aindréas Ó hAodha
2024-06-08 22:44:20 +01:00
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{:highlights []} {:highlights [], :extra {:page 1}}

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- -
- # Adjacency Matrices - # Adjacency Matrices
- What is an **adjacency matrix**? #card - What is an **adjacency matrix**? #card
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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 - 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
- $$a_{i,j} = - $$a_{i,j} =
\begin{cases} \begin{cases}
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\end{cases}$$ \end{cases}$$
- ![image.png](../assets/image_1669209153547_0.png) - ![image.png](../assets/image_1669209153547_0.png)
- ## Properties of the Adjacency Matrix #card - ## Properties of the Adjacency Matrix #card
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- The adjacency matrix of a graph is **symmetric**. - The adjacency matrix of a graph is **symmetric**.
- If $B = A^k$, then $b_{i,j}$ is the number of paths of length $k$ from vertex $i$ to vertex $j$. - If $B = A^k$, then $b_{i,j}$ is the number of paths of length $k$ from vertex $i$ to vertex $j$.
- We can work out if a graph is connected by looking at the eigenvalues of $A$. - We can work out if a graph is connected by looking at the eigenvalues of $A$.
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- What is the **eccentricity of a vertex**? - What is the **eccentricity of a vertex**?
- The **eccentricity of a vertex** is the greatest distance between that vertex & any other vertex in the graph. - The **eccentricity of a vertex** is the greatest distance between that vertex & any other vertex in the graph.
- What is the **radius of a graph**? #card - What is the **radius of a graph**? #card
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- The **radius of a graph** is the minimum eccentricity of any vertex. - The **radius of a graph** is the minimum eccentricity of any vertex.
- What is the **diameter of a graph**? - What is the **diameter of a graph**?
- The **diameter of a graph** is the maximum eccentricity of any vertex. - The **diameter of a graph** is the maximum eccentricity of any vertex.

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file:: [MA284-Week11_1668603812290_0.pdf](../assets/MA284-Week11_1668603812290_0.pdf)
file-path:: ../assets/MA284-Week11_1668603812290_0.pdf

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