Reversed Letter Word Formations and Graph Theory

Take the following words:

                    stop

                    star

Reverse them, and you get:

                          pots

                          rats

Of course, there are many such words (I believe) in English that, when reversed, yield another English word.  But how many are there? What is the percentage of the total lexicon that exhibits this property? Is this percentage growing?  How does it compare to other languages?  I don't have the answer to the latter question, but I would like to research this aspect of human languages one day.  One definite answer comes to mind immediately: reversed letter word formation does NOT increase the size of the English vocabulary.  The reason is obvious, but it highlights an unusual ability to create a directed graphical map of the language.  If that is the case, we can ask whether this directed graphical mapping leads to further mappings?  Let me clarify this idea. We can rearrange the reversed words, i.e., make an anagram, then reverse it again, to map to another English word.  The word could be the same as the original word.  For example, take the word stop above; it can be anagrammed to be tops. So, its directed graph, which would be the map below:

stop

pots

tops

The graph would be a triangle with 3 nodes and 3 edges

What is needed is to enumerate the subset of words with this quality, and then a second test of reversibility to locate new words after they are anagrammed.  And yes, of course, further operations of this type to see if this directed graph can be extended.  This is a simple, straightforward process that any high-level programming language like Python, could perform given an input dataset.  What is not clear is where in graph theory mathematics would lead.  We might find that a rich network of reversed/anagrammed/reversed words can be made to connect English words.

Well, this is exactly what I've begun to do.  Included in this article is an Excel sheet with a VBA macro that reverses common English words. I have made a sample of approximately 50,000 words, most of which are 3 to 5 letters long, and there are some amazing results.  Above is an image of the directed graph using reversed words. A black and white image of lines and dots Description automatically generated

Here is a much better image after some processing and mapping of directed word pairs. Maybe not as aesthetically attractive, but more informative.  Look closely at the image below; you can see that matching pairs of reversed words are shown in purple.  They represent 389 words. That is, words that reverse to form other English words, and the remaining grey nodes with edges are non-English word mappings.

A green spiral on a white background Description automatically generated

I have uploaded the source data for anyone interested in reviewing it.  There is a VBA Excel sheet that was used to generate the source data.  The Gephi program was used to create the images above.

Below is the link for downloading.

Reversed English Words

Permutations, Graph Theory, and Reversed Words

This idea becomes even more fascinating if we compare combinations versus permutations of these words. First, let's set the stage with a little background on combinatoric formulations.

   Permutations of sets of objects can be calculated using the formula = n! / (n-r), where N = set of objects and r = a subset sample of N. The arrangement of subsets is important. A permutation is a distinct set pairing operation, which differs from a combination by the omission of the variable n in the formula. This omission significantly reduces the number of ordered pairs.

   Combinations of sets of objects can be calculated using the formula = n! /n! (n-r), where n = set of objects and r = a subset sample of n. The bold n indicates the factor difference in the formulation.

Let's calculate P(4,2) and C(4,2).

P(4,2) = 4!/(4!-2!) =12 and C(4,2) = 4!/2!(4-2)! = 6

There are more permutations of a given number pair set than combinations of the same number pair set. This is the key to our next step. What is that step? It is crying out to be heard! How many more reversed words would be generated if we allowed the reversed words to be rearranged based on their permutation structure? The number sets would grow immensely if we increased r in permutations. We don't know if these subsets would generate English words; it is randomness that would determine the number of reversed English words that would be generated. Please note that the set could explode. I have an example to illustrate that the remapped set would not. So, take the word reward. Calculate P(6,4) and the result is 360. Below is a partial listing:

This is an abbreviation of the total list, and yes, you guessed it, there are a few English words in there; I have marked them in upper case. The subset of permutated English words of REWARD is small. This is for a 6-letter word, I ask, think of a 7- or 8-letter word. If we take the new word formed and reverse it again, we can find perhaps more of these reversed word pairs. This is the mind-bending part. Right before us is the possibility to find new connections within the linguistic structure of this language (and others by association), through combinatorics and graph theory. I suspect the lack of diacritical characters in English increases the probability of finding reversed word pairs. However, I can't offer a probabilistic proof of this guess.  I could be wrong on this point. Another reason this is becoming so interesting to me.

Robleh Wais 11/5/2023