I was playing around with the chromatic polynomial (denoted here by $\chi_G(x)$) and I have made the following conjecture.

Let $(G_n)_{n \ge 1}$ be a sequence of graphs with $v(G_n) \to \infty$ ($v(G_n)$ denotes the number of vertices of $G_n$) and $e(G_n) \to \infty$ ($e(G_n)$ denotes the number of edges of $G_n$).

For each $x \neq 0$, let us define the following transform of the chromatic polynomial of $G_n$ $$ \psi_{G_n}(x) = \frac{x^{v(G_n)}}{e(G_n)^{v(G_n)}} \chi_{G_n}\left( \frac{e(G_n)}{x} \right). $$

The conjecture is that for each fixed real number $x \neq 0$, we have $\psi_{G_n}(x) \to \exp(-x)$ as $n$ goes to infinity.

I have checked the conjecture for a few sequences of graphs: for example, $G_n$ being the complete graph $K_n$, for $G_n$ being a tree on $n$ vertices and for $G_n$ being a collection of $n$ independent edges (a matching on $2n$ vertices).

Does anyone know if this is well-known?

PS: I am not sure if the conditions on $v(G_n)$ and $e(G_n)$ are the right one. Any comments on this are welcome as well.

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