Two Results on Real Zeros of Chromatic Polynomials

The largest non-integer real zero of chromatic polynomials of graphs with fixed order

It is easy to verify that the chromatic polynomial of a graph with order at most 4 has no non-integer real zeros, and there exists only one 5-vertex graph having a non-integer real chromatic root. This paper shows that, for 66 n6 8 and n¿ 9, the largest non-integer real zeros of chromatic polynomials of graphs with order n are n − 4 + =6 − 2= , where = ( 108 + 12 √ 93 )1=3 , and ( n− 1 +√(n− 3)...

Chromatic polynomials of random graphs

Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009New J. Phys. 11 023001) nowmake it possible to compute chromatic polynomia...

Zeros of adjoint polynomials of paths and cycles

The chromatic polynomial of a simple graph G with n > 0 vertices is a polynomial Σk=1α(G, k)(x)k of degree n, where (x)k = x(x− 1) . . . (x− k+1) and α(G, k) is real for all k. The adjoint polynomial of G is defined to be Σk=1α(G, k)μ , where G is the complement of G. We find the zeros of the adjoint polynomials of paths and cycles.

On the average number of sharp crossings of certain Gaussian random polynomials

A zero-free interval for chromatic polynomials

Woodall, D.R., A zero-free interval for chromatic polynomials, Discrete Mathematics 101 (1992) 333-341. It is proved that, for a wide class of near-triangulations of the plane, the chromatic polynomial has no zeros between 2 and 2.5. Together with a previously known result, this shows that the zero of the chromatic polynomial of the octahedron at 2.546602. . . is the smallest non-integer real z...