Add 2026.07.15 session

notes-and-examples/2026.07.15/graphs-intro.md

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+## Fundamentals
+
+- A *node* or *vertex* contains a data point
+- An *edge* is what connects one node to another
+
+A graph is just a collection of vertices and edges.
+
+> [!QUESTION] What is a real-life example of a simple graph with only vertices and edges?
+
+Some additional properties we can put on a graph:
+
+- A *weight* is a numerical value that can be assigned to an edge
+- We can also assign a *direction* to an edge, such that it only points from one node to another, not the other way around
+
+We can, of course, combine both of these properties too.
+
+> [!QUESTION] What's something we can model with weights in a graph? What about with directional edges?
+
+Going forward, we'll use $V$ to represent the number of vertices in a graph, and $E$ to represent the number of edges.
+
+## Representing a graph
+
+The *adjacency list* stores a list of connected vertices for each node, and it can fit in a dictionary or hash map structure.
+
+```
+1 -> 2, 4
+2 -> 1, 3
+3 -> 2
+4 -> 1
+```
+
+> [!QUESTION] How would you draw out this graph?
+
+> [!QUESTION] What would this mapping look like if we wanted to add weights? What about directional edges?
+
+The *adjacency matrix* is a 2D array where each position `arr[x][y]` represents an edge, and `x` and `y` each represent a node.
+
+```
+[
+    [0, 1, 0, 1],
+    [1, 0, 1, 0],
+    [0, 1, 0, 0],
+    [1, 0, 0, 0]
+]
+```
+
+> [!QUESTION] How would you draw out this graph? What would it look like as an adjacency list?
+
+> [!QUESTION] How do we add weights and/or directional edges to this graph?
+
+## Categorizing graphs
+
+We say a graph is *directed and acyclic*, or a *directed acyclic graph (DAG)*, if there are no cycles formed using the directional edges.
+
+> [!QUESTION] What's something we can model with a DAG?
+
+> [!QUESTION] What's another data structure we went over that also classifies as a DAG?
+
+A graph is *dense* if $E$ is closer to $V^2$, and *sparse* if $E$ is closer to $V$.
+
+> [!QUESTION] What's the implication for the graph if $E$ is closer to $V^2$, i.e. how is it different compared to a sparse graph?
+
+> [!QUESTION] What's the maximum number of edges we can have in a graph with no self-loops, relative to the number of vertices $V$?
+
+A graph can contain *self-loops* (a loop from a vertex to itself). A graph can also have *multiple edges* going from one vertex to another.
+
+> [!QUESTION] How do we represent a self-loop using an adjacency matrix?
+
+Finally, a graph is *connected* if every vertex is reachable from every other vertex.
+

notes-and-examples/2026.07.15/overview.md

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+## Outline
+
+- Go over the homework
+- Go over the basics of graphs, see [Introduction to graphs](./graphs-intro.md)
+
+![Graphs introduction whiteboard](./graphs.png)
+
+To open the `.excalidraw` file:
+
+- Launch https://excalidraw.com
+- Click the top left menu, then select "Open"
+- Select the `.excalidraw` file from the file picker
+
+## Assignment
+
+- Try to complete the red-black implementation from last week
+- Complete the exercise for the 2D array (see the picture)
+