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We know that $DS$ graphs are such connected graphs that determinable by their adjacency spectrum.

Suppose $DS(n)$ and $G(n)$ show the number of $DS$ graphs and all graphs with $n$ vertices,respectively.

$1)$ Do we have any good approximation for $DS(n)$(even if $n$ be sufficiently large)?

$2)$ what is the behavior of $‎\alpha$, if we have:

‎ $lim (DS(n)‎‎/ (G(n)-DS(n))^\alpha=c‎\neq‎0)$

$n ‎\rightarrow‎‎ ‎\infty‎$

Is there any new survey about DS graphs, after 2010?

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Haemers conjectures that the proportion of graphs characterized by their spectrum goes to 1 as $n$ increases. And I am wondering how often you expect the topic to be surveyed. –  Chris Godsil Dec 26 '11 at 15:21
    
When the time of growing of tulips comes, in every land, tulips will grow."BOLYAI" I designed some problems that are interesting for me and I have some plan for solving them. I searched a lot about them, but I didn't found exact answer. Many of my personal problem, as I see here, worked before. What do you think Dear Godsil? Is it good or bad? Please give me reference about this conjecture. Thanks –  Shahrooz Dec 26 '11 at 19:09
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1 Answer

up vote 6 down vote accepted

As far as I know, the computation of these values up to 11 vertices by van Dam and Haemers is still the best result. No asymptotics are known.

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For us ignoramus, what ARE DS graphs? –  Igor Rivin Dec 26 '11 at 12:02
    
Shahrooz's first line is a definition, even though it doesn't look like one. A graph is DS if it is the only graph whose adjacency matrix has that characteristic polynomial. This is not standard terminology. –  Brendan McKay Dec 26 '11 at 14:24
    
Dear McKay, I found a paper with name "Cospectral graphs on 12 vertices"(A. E. Brouwer, E. Spence). –  Shahrooz Dec 26 '11 at 19:43
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