The hottest Graph Theory Substack posts right now

And their main takeaways
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How to turn a number into a rooted forest

A Piece of the Pi: mathematics explained • 90 implied HN points • 01 Mar 26
🔬 Science Graph Theory
  1. Matula arborification is a recursive recipe that turns any positive integer into a rooted forest: 1 is the empty forest, 2 is a single node, primes become trees by attaching a new root to the forest of their index, and composites are represented by juxtaposing the trees of their prime factors.
  2. This correspondence is useful in number theory and combinatorics — it can help prove relationships between primes and encodes integer sequences (for example the primeth sequence appears as vertical chains of trees).
  3. The idea also has practical applications in chemistry for canonically labeling alkane structures (with valence limits ruling out some forests), and there are online tools that generate and visualize Matula trees for given integers.

Surfaces and sequence graphs

A Piece of the Pi: mathematics explained • 66 implied HN points • 31 Jan 26
🔬 Science Graph Theory
  1. You can build a graph by placing n vertices in a cycle and linking them according to the rank order of the first n terms of a real sequence, and as n grows these sequence graphs reveal striking geometric patterns.
  2. Graphs coming from the Kronecker sequence (multiples of the golden ratio mod 1) can be drawn on a torus without crossings, typically after removing the edge from n−1 to 0.
  3. Graphs from the van der Corput sequence embed into the Chamanara surface — a highly singular, infinite‑handle (“Loch Ness monster”) surface made by identifying shrinking boundary segments of a square — and finite approximations avoid the worst singularities so they can be visualized.

Colouring identical cuboids

A Piece of the Pi: mathematics explained • 30 implied HN points • 30 Dec 25
🔬 Science Graph Theory
  1. The number of colours needed depends on the cuboids' sizes and orientations; even with all pieces aligned, some stacks need five colours (2×2×1), others need four (3×1×1) or three (2×1×1).
  2. Letting identical cuboids meet at right angles in the same plane can raise the colour count — for example, 2×1×1 blocks in mixed planar orientations can require five colours.
  3. Allowing arbitrary orientations makes the problem harder: some constructions with 4×1×1 cuboids force at least six colours, there is a proven upper bound of 12 in that case, and it’s unknown whether six is the true maximum overall.

Edge 455: Building Smaller Foundation Models Using Graph-Based Distillation

TheSequence • 105 implied HN points • 10 Dec 24
🕹 Technology Graph Theory
  1. Graph-based distillation helps smaller models learn better by using the connections between data points. Instead of just focusing on individual data, it looks at how they relate to one another.
  2. This technique uses attention networks to improve how student models understand data, making them more effective in learning.
  3. There’s a new framework called Hugging Face Autotrain that allows for easier training of foundation models without needing too much coding knowledge.
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Graphing in the blue skies

Conspirador Norteño • 36 implied HN points • 27 Jun 25
🕹 Technology Graph Theory
  1. Repost network graphs are helpful to understand how ideas spread on social media, especially on platforms like Bluesky. You can visualize how hashtags or posts gain popularity and interaction.
  2. You can create these graphs from datasets of reposts and original posts, using Python to handle the data. This allows researchers to analyze which accounts are most influential in sharing content.
  3. Different types of conversations on social media can create unique patterns in these graphs. For instance, debates might lead to clusters of accounts, while friendly interactions could show a more unified graph with fewer divisions.

The Math Behind Your Commute

The Palindrome • 4 implied HN points • 05 Dec 25
🕹 Technology Graph Theory
  1. BFS explores outward from a start node in layers, so it finds shortest paths by edge count and builds a BFS tree that groups nodes by distance.
  2. DFS follows one path as far as it can and backtracks when needed, producing a narrow, deep DFS tree. Any non-tree edge in DFS must connect a node to one of its ancestors or descendants.
  3. A connected component is the set of nodes reachable from a start node, and both BFS and DFS can find this component to decide if two nodes are connected. More generally, you can discover a component by repeatedly adding neighbors of the current set until no new nodes appear.

k-Core Decomposition

Graphs For Science • 52 implied HN points • 24 Feb 24
🔬 Science Graph Theory
  1. k-Core Decomposition is a way to explore the structure of networks by identifying the largest subgraph where every node has a specified minimum degree.
  2. The k-Core Decomposition algorithm involves recursively removing nodes with degrees lower than a specified threshold to reveal the k-core and k-shell structure of a graph.
  3. The degree of a node in a k-core doesn't have an upper limit, providing unique insights into network connectivity beyond traditional degree-based analysis.

The countable random graph

Infinitely More • 10 implied HN points • 10 Feb 24
🔬 Science Graph Theory
  1. A countable random graph is a graph where you flip a coin to decide the edges between vertices in an infinite set, and the result is the same graph almost every time.
  2. Graph theory is a complex subject with beautiful theorems, and different notions of graphs exist, such as directed graphs and simple graphs.
  3. In mathematics, there are variations in graph definitions, such as allowing reflexivity or multiple edges, but in simpler contexts, graphs are typically referred to as simple graphs.

The Computer Scientist's Guide to Graph Theory, ep. 02

The Palindrome • 3 implied HN points • 13 Dec 23
🔬 Science Graph Theory
  1. Matching problems can be modeled using bipartite graphs where no edges go between vertices of the same type.
  2. In graph theory, a full matching of one partition of a bipartite graph implies that every vertex in that partition has at least as many neighbors in the other partition.
  3. Hall's theorem provides a necessary and sufficient condition for determining the existence of a full matching in a bipartite graph.