fixes. up to ring graph

Author: mcloughlan

docs/algorithms.md

@@ -658,7 +658,7 @@ When stuck at:
 
 Basically, "*Calling a function inside of itself*"
 
-<img src="images/Pasted image 20220324205607.png" alt="Pasted image 20220324205607">
+![Recursive Serpinski triangle](images/Pasted image 20220324205607.png){width=200}
 
 Requirements:
 
@@ -740,7 +740,7 @@ Where:
 
 	Also, another geometric time complexity is a decision tree, where it expands in $2^n$ time
 
-	<img src="images/Pasted image 20220803103031.png" alt="Pasted image 20220803103031">
+	<img src="images/Pasted image 20220803103031.png" alt="Pasted image 20220803103031" width=300px>
 
 	Another example is $T(n) = T(n/3) + T(2n/3) + n$. I just need to note that:
 	- Asymptotic height of tree is longest term -> $T(2n/3)$
@@ -763,6 +763,12 @@ $${PR(A)={\frac {PR(B)}{2}}+{\frac {PR(C)}{1}}+{\frac {PR(D)}{3}}}$$
 
 !!! info
 
+	Cool historical and general technical explanation. Only a few minutes:
+
+	<iframe width="560" height="315" src="https://www.youtube.com/embed/KZeIEiBrT_w?si=9LRXiCsQumdq3cWe&amp;start=1147" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
+
+	Technical video:
+
 	<iframe width="560" height="315" src="https://www.youtube.com/embed/qxEkY8OScYY?si=rg-kzamjXyxP05DI" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
 
 ##### Damping factor
@@ -842,7 +848,7 @@ State the proposition, say "the proof is by contradiction", then contradict your
 
 ### Axioms
 
-<img src="images/Pasted image 20220905092544.png" alt="Pasted image 20220905092544">
+<img src="images/Pasted image 20220905092544.png" alt="Pasted image 20220905092544" width=300px>
 
 - As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy
 
@@ -1202,11 +1208,11 @@ Examples:
 <img src="images/Pasted image 20220820151008.png" alt="Pasted image 20220820151008" width="300">
 
 - Better algorithms are still possible
-	- Better algorithms will not provide an improvement detectable by “Big O”
+	- Better algorithms will not provide an improvement detectable by "Big O"
 
 ### Bisection solve
 
-<img src="images/Pasted image 20220713093654.png" alt="Pasted image 20220713093654">
+<img src="images/Pasted image 20220713093654.png" alt="Pasted image 20220713093654" width=400px>
 
 Given lower bound $a$ and upper bound $b$, calculate the mid (average) value $c$ and sub into $f(x)$. Then adjust the bounds $a, b$ to move closer to the value closer to $0$.
 
@@ -1454,7 +1460,7 @@ As you can see, they are also very sensitive to input 😢
 
 Forward propagation is the initial phase of data processing in a neural network. Here, input data is fed into the network and passed through various layers. Each neuron in these layers processes the input and passes it to the next layer, ultimately leading to the output layer. This process is linear and straightforward, moving in one direction: from input to output.
 
-<img src="images/Pasted image 20240910154731.png" alt="Pasted image 20240910154731">
+<img src="images/Pasted image 20240910154731.png" alt="Pasted image 20240910154731" width=400px>
 
 [source](https://stackademic.com/blog/the-difference-between-back-propagation-and-forward-propagation-in-deep-learning-2b2248e6d00c)
 

docs/graphs.md

@@ -52,11 +52,11 @@ The neighbourhood of a vertex $v$ is the set of vertices adjacent to $v$.
 
 #### Order
 
-Number of [vertices](#nodes) a graph has
+The number of [vertices](#nodes) in a graph.
 
 ### Edges
 
-Edges are the lines in-between [Nodes](#nodes) that display a connection between nodes.
+Edges are the lines in-between [nodes](#nodes) that display a connection between nodes.
 Edges represent connections, relations or any logical path.
 
 - We denote the edge set of a graph $G$ by $E(G)$
@@ -77,7 +77,7 @@ If the vertex $u$ is connected to $e$; $e$ is an edge incident to $u$
 
 A loop is an edge that has the same starting and ending [node](#nodes). See node 1 below:
 
-<img src="images/Pasted image 20220222181219.png" alt="Pasted image 20220222181219">
+<img src="images/Pasted image 20220222181219.png" alt="Pasted image 20220222181219" width=300px>
 
 ### Paths
 
@@ -89,27 +89,28 @@ A path can contain many edges, or a single. it depends on what nodes are being s
 
 !!! note
 
-	- [Euler paths](#euler-path) and [Euler circuits](#euler-circuit):
-		- Can pass through [nodes](#nodes) more than once.
-		- Don't exist if a graph has more than two vertices of odd degree.
-		- Exists if all vertices of a graph have even degree.
-		- Exists if a connected graph has exactly two odd vertices. The starting point must be one of the odd vertices and the ending point will be the other of the odd vertices.
-
-	- [Hamiltonian Paths](#hamiltonian-path)
-		- Don't have to traverse every [edge](#edges)
+	- Check out [Euler paths](#euler-path) and [Euler circuits](#euler-circuit):
+	- Check out [Hamiltonian Paths](#hamiltonian-path)
 
 
 #### Euler Path
 
 A [path](#paths) that uses each [edge](#edges) once only without returning to starting [vertex](#nodes) is an Euler path.
 
-- Uses each **edge** once
+![euler path](images/euler-path.png)
+
 - Doesn't return to starting vertex
+- Can pass through [nodes](#nodes) more than once.
+- Don't exist if a graph has more than two vertices of odd degree.
+- Exists if all vertices of a graph have even degree.
+- Exists if a connected graph has exactly two odd vertices. The starting point must be one of the odd vertices and the ending point will be the other of the odd vertices.
 
 #### Euler circuit
 
 A [path](#paths) that uses each [edge](#edges) once only and returns to starting [vertex](#nodes) is an Euler circuit.
 
+![euler circuit](images/euler-circuit.png)
+
 If an undirected graph $G$ is connected and every vertex  (not isolated) in $G$ has an even degree, then $G$ has an Euler circuit.
 
 - Uses each **edge** once
@@ -119,15 +120,24 @@ If an undirected graph $G$ is connected and every vertex  (not isolated) in $G$
 
 A [path](#paths) that passes every [vertex](#nodes) exactly once without returning to starting [vertex](#nodes) is an Hamiltonian path.
 
+![hamiltonian path](images/hamiltonian-path.png)
+
 - Uses every **node** once
 - Doesn't return to starting node
+- Don't have to traverse every [edge](#edges)
 
 #### Hamiltonian Circuit
 
 !!! note
 
 	This is used very often in this course. See [Travelling salesmen problem](algorithms.md#travelling-salesmen-problem)
 
+!!! important
+
+	A [cycle](#cycle) and a [circuit](#circuit) are different. But the properties of both Hamiltonian cycles and Hamiltonian circuits are the same. Hence they reffer to the same thing.
+
+![Hamiltonian circuit](images/hamiltonian-circuit.png){width=300px}
+
 A [path](#paths) that passes every [vertex](#nodes) exactly once and returns to the starting [vertex](#nodes) is an Hamiltonian path.
 
 - Uses every **node** once
@@ -145,16 +155,28 @@ Is there a path from $A$ to $B$ ?
 
 The transitive closure of a graph is a graph which contains an edge between $A$ and $B$ whenever there is a directed path from $A$ to $B$. In other words, to generate the transitive closure every path in the graph is directly added as an additional edge.
 
-<img src="images/Pasted image 20220321094104.png" alt="Pasted image 20220321094104">
+<img src="images/Pasted image 20220321094104.png" alt="Pasted image 20220321094104" width=300px>
 
 A graph $G^*$ which is the transitive closure of $G$ will have a directed edge to every node it can traverse to.
 
+
+### Undirected Graphs
+
+![undirected graph](images/undirected-graph.png){width=200}
+
+Undirected graphs have [edges](#edges) that can be travelled along in any direction.
+
 ### Directed Graph
 
 Also known as a digraph.
-A digraph is an ordered pair [G = (V, E)](#graph) of [edges](#edges) $(x,y)$ of arrows "from $x$ to $y$".
+
+![directed graph](images/directed-graph.png){width=200}
+
+Bassically just the same as a regular [undirected graph](#undirected-graphs), but the edges have arrows (direction). 
+Formally, a digraph is an ordered pair [G = (V, E)](#graph) of [edges](#edges) $(x,y)$ of arrows "from $x$ to $y$".
 
 Where:
+
 - $x$:
 	- Is the tail
 	- Is the direct predecessor of $y$
@@ -168,11 +190,7 @@ If a [path](#paths) leads from $x$ to $y$, then $y$ is said to be a successor of
 
 A directed graph $G$ is strongly connected if there is a directed path from every vertex to every other vertex in $G$.
 
-<img src="images/Pasted image 20220222220031.png" alt="Pasted image 20220222220031">
-
-### Undirected Graphs
-
-Undirected graphs have [edges](#edges) that can be travelled along in any direction.
+<img src="images/Pasted image 20220222220031.png" alt="Pasted image 20220222220031" width=500px>
 
 ### Dag / DAGS
 
@@ -232,7 +250,7 @@ Similarity = 0 if $X \neq Y$
 
 !!! warning
 
-	writing this in 2025. Never seen cosine simularity anywhere. dw about it
+	I have never seen cosine simularity anywhere. Don't worry too much about it
 
 Best example in the world:
 
@@ -248,7 +266,7 @@ For unweighted graphs, each edge has a nominal distance of 1.
 
 #### Euclidian distance
 
-Is also known as simply distance
+The most commonly understood distance metric
 
 Euclidean Distance is the straight line distance between two entities. Which is worked out using **Pythagoras Theorem**
 
@@ -262,7 +280,8 @@ In a simple way of saying it is the total sum of the difference between the x-co
 
 The [heuristic function](#heuristic-functions) for distance calculation
 
-$$f(n) = |\text{cell}_{row} – \text{goal}_{row}| + |\text{cell}_{col} – \text{goal}_{col}|$$
+{# idk why but the latex didn't render for this. So just rasterized it instead #}
+![distance-heuristic](images/distance-heuristic.png)
 
 Which is the absolute cell distance in the $x$ direction plus the absolute cell distance in the $y$ direction. The absolute value is always positive.
 
@@ -279,9 +298,9 @@ Also known as a topological ordering
 
 In computer science a **topological sort** or **topological ordering** of a [Directed Graph](#directed-graph) is a linear ordering of its [vertices](#nodes) such that for every directed edge _uv_ from vertex _u_ to vertex _v_, _u_ comes before _v_ in the ordering.
 
-A topological ordering is possible iff the graph has no directed cycles, that is, if it is a directed acyclic graph ([Dag DAGS](#dag-dags)).
+A topological ordering is possible iff the graph has no directed cycles, that is, if it is a directed acyclic graph ([Dag - DAGS](#dag-dags)).
 
-<img src="images/Pasted image 20220310113016.png" alt="Pasted image 20220310113016">
+<img src="images/Pasted image 20220310113016.png" alt="Pasted image 20220310113016" width=400px>
 
 The graph shown above has many valid topological sorts, including:
 
@@ -302,7 +321,7 @@ This can be calculated through [Kahn's Algorithm](#kahns-algorithm)
 
 A [Topological sort](#topological-sort) algorithm, first described by **Kahn** (1962), works by choosing vertices in the same order as the eventual topological sort. First, find a list of "start nodes" which have no incoming edges and insert them into a set $S$; at least one such node must exist in a non-empty acyclic graph. Then:
 
-```js
+```js title="Kahn's Algorithm in psuedocode"
 L ← Empty list that will contain the sorted elements
 S ← Set of all nodes with no incoming edges
 
@@ -319,11 +338,13 @@ else
     return L (a topologically sorted order)
 ```
 
-### Graph Diameter
+### Graph Diameter and radius
 
 !!! note
 
-	Note that the **diameter** is the max graph ecentricity. The radius of the graoh requires a center point to be defined
+	Note that the **diameter** is the max graph ecentricity. The radius of the graph requires a center point to be defined
+
+![graph radius and diameter example](images/radius-diamater-example1.png)
 
 The longest shortest path between any two nodes counted by edge and weights.
 
@@ -337,11 +358,12 @@ The examples below have a diameter of 2 (2 edges to traverse to get from a to b)
 
 <img src="images/Pasted image 20220303114136.png" alt="Pasted image 20220303114136">
 
-<img src="images/Pasted image 20220919214414.png" alt="Pasted image 20220919214414">
+Example:
+
+<img src="images/Pasted image 20220919214414.png" alt="Pasted image 20220919214414" width=400px>
 
 *Note that the radius and diameter is counted by the number of edges
 
----
 
 ## Network graphs
 
@@ -374,13 +396,13 @@ We can then obtain $H$ from $G$ by deleting edges and or vertices from  $G$.
 
 #### Simple graph
 
-A simple graph, also called a strict graph is **an unweighted, undirected graph containing no graph loops or multiple edges**.
+A simple graph, also called a strict graph is **an unweighted, [undirected graph](#undirected-graphs) containing no graph [loops](#loops) or multiple edges**.
 
-A simple graph may be either connected or disconnected. Unless stated otherwise, the unqualified term "graph" usually refers to a simple graph.
+A simple graph may be either connected or [disconnected](#disconnected-graph). Unless stated otherwise, the unqualified term "graph" usually refers to a simple graph.
 
 <img src="images/Pasted image 20220222144711.png" alt="Pasted image 20220222144711">
 
-For simple [**connected graphs**](#connected-graph) the amount of edges you can have are striclty bounded by:
+For simple [**connected graphs**](#connected-graph) the amount of edges you can have are striclty bounded. The number of edges will always be between the arrangement of a [tree](#trees) or a [complete graph](#complete-graphs):
 
 $$
 \overset{\text{(trees)}}{(|V|-1)}
@@ -391,20 +413,24 @@ $$
 
 #### Multigraph
 
+![multigraph](images/multigraph.png){width=300px}
+
 A graph that can have multiple edges between the same pair of nodes. In a road network this could, for example, be used to represent different routes with the same start and end point.
 
 #### Connected Graph
 
+![connected vs disconnected graphs](images/connected-vs-disconnected-graphs.png){width=500px}
+
 A graph where all [nodes](#nodes) are connected
 
 ##### Ring Graph
 
-Also known as a Cycle Graph
+Also known as a [Cycle](#cycle) Graph
 A graph that forms a ring
 
-- All nodes have a degree of 2
+- All nodes have a [degree](#degree) of 2
 
-<img src="images/Pasted image 20220222220633.png" alt="Pasted image 20220222220633">
+<img src="images/Pasted image 20220222220633.png" alt="Pasted image 20220222220633" width=300px>
 
 #### Disconnected Graph