Let $T\_G(x,y)$ denote the Tutte polynomial of a graph. Of course we may have $T\_G(x,y) = T\_H(x,y)$ for $G$ and $H$ non-isomorphic graphs.


Now let $c(G)$ denote the *cone graph* of $G$, i.e., the graph obtained from $G$ by adding a new vertex connected by an edge to every vertex of the original graph $G$. Let $c^{n}(G)$ denote the graph obtained by coning over $G$ $n$ times.


Note that we can have $T\_G(x,y) = T\_H(x,y)$ but $T\_{c(G)}(x,y) \neq T\_{c(H)}(x,y)$. For instance, take $G = K\_{1,3}$ the star with three edges, and $H = P\_3$ the path with three edges. Then $T\_G(x,y) = T\_H(x,y)$ because all trees on the same number of edges have the same Tutte polynomial, but $T\_{c(G)}(x,y) \neq T\_{c(H)}(x,y)$ because for instance $c(G)$ has $20$ spanning trees and $c(H)$ has $21$ spanning trees (and the number of spanning trees of any connected graph is the Tutte polynomial evaluated at $(1,1)$.)


**Question**: Is it the case that if $G$ and $H$ are non-isomorphic graphs then $T\_{c^n(G)}(x,y) \neq T\_{c^n(H)}(x,y)$ for some $n$? If so, are there some effective bounds on this $n$?



Does the Tutte polynomial of iterated cone graphs detect isomorphism?
I don't think that's true - I think the idea from [this](http://arxiv.org/abs/1306.0864) paper works out.


Let $G$ be a simple graph and let $H$ be a spanning subgraph of $G$ having connected components of order $h\_1 \geq h\_2 \geq \cdots \geq h\_k.$ We say that $(|E(H)|,h\_1,h\_2, \ldots, h\_k)$ is a s*ubgraph description* of $H.$ Let now $s(G)$ be the lexicographically sorted tuple of subgraph descriptions for every subgraph of $G.$ We call $s(G)$ the *subgraph sequence* of $G.$


Notice that if $s(H) = s(G)$ then $s(c(H)) = s(c(G)).$


Now there are two non-isomorphic graphs having the same subgraph sequence and you can find such an example on 8 vertices in the linked paper.


But the subgraph sequence of a graph uniquely determines its Tutte polynomial so such graphs should be a counterexample.


