Word Shapes
Comparing Words by the LetterAdjacency Graphs.
This video by John Turner has a fun idea: Looking at which words are ‘shaped’ the same. One of the ways he defines a word’s shape is via its graph^{1} of letter adjacencies. For example, “baboon” and “refers” have the same graph shape because the network of connections between adjacent letters is similar.
Unfortunately, despite using a graphing library called Scott to compute canonical representations of each word’s graph, what Turner has calculated doesn’t seem to actually be (just) about graph isomorphism. It also takes into account the position of letters around the “letter wheel”. I found this unsatisfyingly restrictive.
Looking at just the networks of letter adjacency, “baboon” should be similar not just to words like “refers”, but also to words like “cats” and “wooly”.
The video made me curious what the results would look like when looking at just the graph of adjcencies between letters in a word. (Henceforce “the word’s graph” for short.)
This post partially answers that question
using the enable1
dictionary of words from this Github Repo
Which Small Graphs are Missing?
4 or fewer nodes
If we’re looking at words with four or fewer distinct letters, there are only a few ‘shapes’ such a word graph could have. (20, to be precise.) And every one of them has at least one corresponding word.
In fact, all of them except for the K4 complete graph have multiple words. K4 only has one, and it’s a bit iffy. It’s “gensengs”, which is the plural of an alternate spelling of “ginseng”.
Should that count? well, it’s in the dictionary I’m using, so 🤷
5 nodes
Of the 21 simple connected graphs with 5 nodes, all are represented except for one. The missing graph is K5, aka the pentatope graph.
No word has this graph. I also checked the larger words and none of them seem to contain this shape as a subgraph either. Spooky!
Here’s a table showing all the graphs with 5 or fewer nodes.
Graph  Example Word  Visualization 

singleton graph  i  
2path  to  
3path  air  
K3 (triangle)  aqua  
paw graph  catch  
4path  fire  
diamond graph  miasma  
square graph  anima  
K4 (tetrahedron)  gensengs  
banner graph  absorb  
fork graph  elixir  
(3,2)tadpole graph  propel  
bull graph  alcohol  
kite graph  calculus  
butterfly graph  tempest  
(4,1)lollipop graph  torturous  
cricket graph  aether  
5path  earth  
dart graph  instant  
5star  kabbalah  
gem graph  seascape  
(2,3)complete bipartite  loyalty  
house graph  automata  
(1,1,3)complete tripartite  attractant  
house X graph  lanolin  
5cycle (pentagon)  exhume  
3dipyramidal  intensities  
5graph 31  nurturant  
5wheel  milliosmols  
K5 (pentatope) 
Names are taken from this page on Biconnected Graphs from Wolfram Mathworld.
6 Nodes
TODO

“Graph” as in “graph theory”, the study of networks and connections. To get a word’s letteradjacency graph: Each letter is a vertex. There is an edge connecting two letters if they show up next to each other in the word. The graphs are simple graphs, meaning we don’t connect a letter to itself (in words like “moon”), nor do we add extra edges when the same adjacency happens multiple times (in words like “donor”). ↩