Several Extreme Coefficients of the Tutte Polynomial of Graphs

Let $$t_{i,j}$$ t i , j be the coefficient of $$x^iy^j$$ x i y j in the Tutte polynomial T ( G ;  x ,  y ) of a connected bridgeless and loopless graph G with order v and size e . It is trivial that $$t_{0,e-v+1}=1$$ t 0 , e - v + 1 = 1 and $$t_{v-1,0}=1$$ t v - 1 , 0 = 1 . In this paper, we obtain expressions for another six extreme coefficients $$t_{i,j}$$ t i , j ’s with $$(i,j)=(0,e-v)$$ ( i , j ) = ( 0 , e - v ) , $$(0,e-v-1)$$ ( 0 , e - v - 1 ) , $$(v-2,0)$$ ( v - 2 , 0 ) , $$(v-3,0)$$ ( v - 3 , 0 ) , $$(1,e-v)$$ ( 1 , e - v ) and $$(v-2,1)$$ ( v - 2 , 1 ) in terms of small substructures of G . We also discuss their duality properties and their specializations to extreme coefficients of the Jones polynomial.