Derivative of Tutte polynomial at -1 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T15:17:24Z http://mathoverflow.net/feeds/question/43363 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43363/derivative-of-tutte-polynomial-at-1 Derivative of Tutte polynomial at -1 Yaroslav Bulatov 2010-10-24T08:25:34Z 2010-10-26T20:07:49Z <p>Let Tutte polynomial on graph with edge-set $E$ be defined as follows</p> <p>$$f(q,v)=\sum_{A\subseteq E} q^{\kappa(A)} v^{|A|}$$</p> <p>Here the sum is over all subgraphs $A$, $\kappa(A)$ is the number of connected components in $A$, $|A|$ is number of edges in $A$.</p> <p>Let $$g(q,v)=\frac{\partial}{\partial v} \log f(q,v)$$ $$h(q,v)=\frac{\partial}{\partial v} f(q,v)$$</p> <p>Is anything known about $g(q,-1)$ or $h(q,-1)$?</p> <p>(<b>Edit 10/26:</b> actually it's the <a href="http://arxiv.org/abs/math/0503607" rel="nofollow">multivariate Tutte polynomial</a> restricted to all $v$'s being equal, relation to regular Tutte polynomial in Jeremy Martin's answer)</p> <p><b>Update 10/24</b></p> <p>$f(q,-1)$ is the number of proper $q$ colorings, $h(q,-1)$ is ???, $g(q,-1)$ is their ratio. Below are tables of values of $h(q,-1)$ for paths, cycles, complete graphs. Columns give graph size, $n=3..6$, rows give $q=2..8$</p> <p><img src="http://yaroslavvb.com/upload/tutte-derivative.png"></p> <p>Mathematica <a href="http://pastebin.com/raw.php?i=9rwkCjGy" rel="nofollow">code</a> to generate this</p> http://mathoverflow.net/questions/43363/derivative-of-tutte-polynomial-at-1/43695#43695 Answer by Jeremy Martin for Derivative of Tutte polynomial at -1 Jeremy Martin 2010-10-26T17:11:00Z 2010-10-26T17:11:00Z <p>This polynomial isn't the usual Tutte polynomial, but it's equivalent. Provided that $G$ is connected (which is probably the case you're interested in, and I'll assume), it looks like $f_G(q,v)=qv^{n-1}T_G(q/v+1,v+1)$, where $n$ is the number of vertices of $G$.</p> <p>One thing that comes to mind is the Crapo beta invariant, which is $(-1)^n\chi_G'(1)$, where $\chi_G(k)=(-1)^{n-1}kT_G(1-k,0)$ is the chromatic polynomial. (E.g., see exercise 22 of lecture 4 of Stanley's notes on hyperplane arrangements: <a href="http://www-math.mit.edu/~rstan/arrangements/arr.html" rel="nofollow">http://www-math.mit.edu/~rstan/arrangements/arr.html</a>.) Your polynomials, especially $g$, might be related to this, but I haven't thought about the details.</p>