Combinatorial interpretations for T_G(1, −1)

Abstract

Let G be a connected and simple graph with vertex set {1, 2, . . . , n+1} and T G (x, y) the Tutte polynomial of G. In this paper, we give combinatorial interpretations for T G (1, −1). In particular, we give the definitions of even spanning tree and left spanning tree. We prove T G (1, −1) is the number of even-left spanning trees of G. We associate a permutation with a spanning forest of G and give the definition of odd G-permutations. We show T G (1, −1) is the number of odd G-permutations. We give a bijection from the set of odd K n+1 -permutations to the set of