I'm looking at a family of graphs that look like this:
$n = 1$
$n = 2$
$n = 3$
So for every $n$ you add another "shell" and connect it with the previous outer shell.
What i'm interested in is if there is a nice way to express the Tutte polynomial of these graphs for every $n$.
Due to the restrictions set by this paper in theorem 1.17 I have a hunch that all the graphs in this family are T-unique and I would like to study the polynomial further. Any help is much appreciated!
The Tutte polynomials for the first three family members look like this: $$T_{G_{n=1}}(x,y) = x^3+x^2+x+y$$
$$T_{G_{n=2}}(x,y) = {x}^{7}+5\,{x}^{6}+15\,{x}^{5}+6\,{x}^{4}y+{y}^{5}+29\,{x}^{4}+24\,{x} ^{3}y+12\,{x}^{2}{y}^{2}+8\,x{y}^{3}+7\,{y}^{4}+40\,{x}^{3}+52\,{x}^{2 }y+39\,x{y}^{2}+20\,{y}^{3}+32\,{x}^{2}+46\,xy+25\,{y}^{2}+11\,x+11\,y $$
$$T_{G_{n=3}}(x,y) = {x}^{11}+9\,{x}^{10}+45\,{x}^{9}+11\,{x}^{8}y+2\,{x}^{4}{y}^{5}+{y}^{9 }+154\,{x}^{8}+88\,{x}^{7}y+28\,{x}^{6}{y}^{2}+16\,{x}^{5}{y}^{3}+14\, {x}^{4}{y}^{4}+8\,{x}^{3}{y}^{5}+8\,{x}^{2}{y}^{6}+8\,x{y}^{7}+11\,{y} ^{8}+396\,{x}^{7}+376\,{x}^{6}y+223\,{x}^{5}{y}^{2}+144\,{x}^{4}{y}^{3 }+108\,{x}^{3}{y}^{4}+92\,{x}^{2}{y}^{5}+78\,x{y}^{6}+58\,{y}^{7}+784 \,{x}^{6}+1030\,{x}^{5}y+833\,{x}^{4}{y}^{2}+624\,{x}^{3}{y}^{3}+484\, {x}^{2}{y}^{4}+356\,x{y}^{5}+192\,{y}^{6}+1203\,{x}^{5}+1964\,{x}^{4}y +1892\,{x}^{3}{y}^{2}+1495\,{x}^{2}{y}^{3}+1000\,x{y}^{4}+441\,{y}^{5} +1395\,{x}^{4}+2580\,{x}^{3}y+2591\,{x}^{2}{y}^{2}+1786\,x{y}^{3}+715 \,{y}^{4}+1148\,{x}^{3}+2149\,{x}^{2}y+1869\,x{y}^{2}+777\,{y}^{3}+589 \,{x}^{2}+948\,xy+498\,{y}^{2}+139\,x+139\,y$$