Several Extreme Coefficients of the Tutte Polynomial of Graphs

Let t i , j be the coefficient of 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 and t v - 1 , 0 = 1 . In this paper, we obtain expressions for another six extreme coefficients t i , j ’s with ( i , j ) = ( 0 , e - v ) , ( 0 , e - v - 1 ) , ( v - 2 , 0 ) , ( v - 3 , 0 ) , ( 1 , e - v ) and ( 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.