Zero-Free Intervals for Flow Polynomials of Near-Cubic Graphs

Let P(G,t) and F(G,t) denote the chromatic and flow polynomials of a graph G. G.D. Birkhoff and D.C. Lewis showed that, if G is a plane near triangulation, then the only zeros of P(G,t) in (−∞,2] are 0, 1 and 2. We will extend their theorem by showing that a stonger result to the dual statement holds for both planar and non-planar graphs: if G is a bridgeless graph with at most one vertex of degree other than three, then the only zeros of F(G,t) in (−∞,α] are 1 and 2, where α ≈ 2.225... is the real zero in (2,3) of the polynomial t4 − 8t3 + 22t2 − 28t + 17. In addition we construct a sequence of ‘near-cubic’ graphs whose flow polynomials have zeros converging to α from above.

• Type: Journal
• Year: 2007
• Where: CPC
• Authors: Bill Jackson
• Keywords: Chromatic, CPC 2007, Graph G, flow Polynomials