A zero-free interval for chromatic polynomials
Discrete Mathematics - Special volume (part two) to mark the centennial of Julius Petersen's “Die theorie der regula¨ren graphs” (“The theory of regular graphs”)
The Zero-Free Intervals for Characteristic Polynomials of Matroids
Combinatorics, Probability and Computing
The Zero-Free Intervals for Chromatic Polynomials of Graphs
Combinatorics, Probability and Computing
Chromatic Roots are Dense in the Whole Complex Plane
Combinatorics, Probability and Computing
European Journal of Combinatorics - Special issue on combinatorics and representation theory
A zero-free interval for flow polynomials of cubic graphs
Journal of Combinatorial Theory Series B
Analytic Combinatorics
Journal of Combinatorial Theory Series B
ACM Transactions on Mathematical Software (TOMS)
| Hi-index | 0.00 |
The number of nowhere zero Z"Q flows on a graph G can be shown to be a polynomial in Q, defining the flow polynomial @F"G(Q). According to Tutte@?s five-flow conjecture, @F"G(5)0 for any bridgeless G. A conjecture by Welsh that @F"G(Q) has no real roots for Q@?(4,~) was recently disproved by Haggard, Pearce and Royle. These authors conjectured the absence of roots for Q@?[5,~). We study the real roots of @F"G(Q) for a family of non-planar cubic graphs known as generalised Petersen graphs G(m,k). We show that the modified conjecture on real flow roots is also false, by exhibiting infinitely many real flow roots Q5 within the class G(nk,k). In particular, we compute explicitly the flow polynomial of G(119,7), showing that it has real roots at Q~5.0000197675 and Q~5.1653424423. We moreover prove that the graph families G(6n,6) and G(7n,7) possess real flow roots that accumulate at Q=5 as n-~ (in the latter case from above and below); and that Q"c(7)~5.2352605291 is an accumulation point of real zeros of the flow polynomials for G(7n,7) as n-~.