I call graphs $G_1$ and $G_2$ distinct iff (i) $G_1$ has a different arrangement¹ of vertices than $G_2$ and (ii) $G_1$ and $G_2$ have the same number of vertices. All other properties of $G_1$ and $G_2$ have no effect on distinguishing them. For example, $G_1$ and $G_2$ may be the same even if the number of edges of the two graphs is different.

Example:

$G_1$	$G_2$	$G_3$

Here, $G_1 = G_2$ but $G_1 \neq G_3 \iff G_2 \neq G_3$.

I'm wondering if there is a way to create a unique correspondence between a graph and a single word (any string of letters, possibly gibberish) of the English alphabet. That is to say, distinct graphs $G_1$ and $G_2$ must necessarily map to a different word. It is preferable that the word has 4-7 characters. Additionally, the word may contain a single integer from 1-9 at the end of it. Hence, for example, $G_1$ could map to the word GLBXR7 and $G_2$ could map to the word MMTR.

I'm looking for an algorithm to map every distinct graph with a unique word. You may assume that every graph has the same number of vertices.

A very crude attempt is to label each vertex of a graph with letters from left to right, top to bottom and concatenate them in that order. However, I don’t think this is sufficient for uniqueness.

Clarifications

I attempt to clarify some of the potentially ambiguous elements of my post.

1. I agree that the notion of a "graph" is not entirely relevant here. I actually chose it because I am working with this sort of information in another project where connectivity matters. Nonetheless, viewing this as a point cloud is perfectly fine; you may edit the tag(s) if you'd like.

2. There doesn’t need to be a bijection between words and graphs. Every unique graph should map to a unique word, but every word need not map to a graph.

3. Since I placed a finite restriction on the length of a word (4 - 7 characters), I attempt to set an upper bound on the number of vertices of a graph.

$$\text{Number of words} = \sum_{k = 4}^7 \left(26^{k-1}\cdot (26 + 9)\right) = 11244509640$$ $$\text{Number of graphs with} \le n \; \text{vertices} = \sum_{k = 1}^n \sum_{i = 1}^k {{k - 1} \choose {i - 1}} = 2^n - 1$$ $$11244509640 = 2^n - 1 \implies \lfloor n\rfloor = 33$$

4. I apologise for not making this point before, but the algorithm should be human computable, that is, possibly something a human can do in their head in about 15 seconds.

¹_{I'm not sure how to perfectly formalise the notion of an arrangement, but a possible necessary and sufficient condition is to consider the number of vertices to the left, right and along the same vertical axis as each vertex of graphs $G_1$ and $G_2$—if any of the values are different, the graphs are distinct.}

posted over 1 year ago

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1y ago

Carefree Explorer‭

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