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Let $G$ be a graph on $n$ vertices. It is quite easy to write out a first-order sentence $\phi$ in the language of graphs such that any graph on $n$ vertices which satisfies $\phi$ is isomorphic to $G$ (by spelling out its adjacency matrix). The length of the sentence in the above method is $O(n^2)$. Sometimes you can do better:

  1. The complete graph on $n$ vertices can be distinguished from other graphs on the vertex set by the formula saying that any two vertices are equal or neighbors, and this formula has constant length.

  2. Similarly, a graph with a clique of size $n-c$ for a constant $c$ can be distinguished via a sentence of constant length.

  3. The complements of these graphs.

I would guess that questions about this length were studied because it seems extremely natural: a graph is the most basic combinatorial object, and you often describe them verbally (although using tools which are more liberal than first-order language, such as "take this graph and attach that graph to it on this vertex"). I can't seem to find any reference to the notion, and was wondering whether anyone can point me in the correct direction. The specific question I had in mind is this: what is the expected value of this length in the $G(n,\frac{1}{2})$ model?

Uri George Peterzil
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  • mmmh, a most efficient language in terms of space would just count or 'enumerate' graphs - see e.g. https://math.stackexchange.com/questions/154941/how-to-calculate-the-number-of-possible-connected-simple-graphs-with-n-labelle?noredirect=1&lq=1 as a starting point --- if instead, you want a language that let's you easily construct the graph from the sentence, the you may want to look at the graph6 standard, e.g., https://stackoverflow.com/questions/44532492/how-does-graph6-format-work – Michael T Mar 18 '25 at 11:40
  • @MichaelT Thank you, but I mean the first-order language of graphs. This doesn't seem related to the links you have provided. – Uri George Peterzil Mar 19 '25 at 07:04
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    ok, seems I have misunderstood and I guess this is not what you mean: https://elbd.sites.uu.nl/wp-content/uploads/sites/108/2017/08/20191008_lamavoc.pdf. Possibly, you may want to consider including a reference or 2 to help readers orient... Many thanks – Michael T Mar 19 '25 at 11:29
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    Based on https://en.wikipedia.org/wiki/Logic_of_graphs#Second_order, I would think that you will need some 'fixed cost' to explain what a graph is (assume you mean undirected, without loops, no multi-edges) and what graph isomorphism is (assume you look at isomorphism classes of unlabelled graphs), then you may have a few exceptional cases as you described but those would be almost non-existent for large random graphs ($n\to\infty$), then you have $n(n-1)/2$ potential edges to describe... – Michael T Mar 19 '25 at 11:43
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    Hi Uri, as far as I understand the case of large random graphs and the sentence terminology in graph logic, I would expect $n(n-1)/2+A$ 'sentences' for $G(n,1/2)$ and large $n$, where $A$ describes the 'fixed cost' definitions (simple graph, isomorphism etc) and assuming undirected graphs (with symmetric adjacency matrix). Put differently, I don't expect an option to go for anything shorter sufficiently often/ almost never. – Michael T Mar 19 '25 at 14:21
  • I really appreciate the answers, thank you! – Uri George Peterzil Mar 19 '25 at 17:24

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