Let P(G;) denote the chromatic polynomial of a graph G. A graph G is chromatically unique if G ∼ = H for any graph H such that P(H;) = P(G;). This note corrects an error in the proof of the chromatic uniqueness of certain 2-connected graphs with n vertices and n + 3 edges.

In this paper we study various extremal problems related to some combinatorially defined graph polynomials such as matching polynomial, chromatic polynomial, Laplacian polynomial. It will turn out that many problems attain its extremal value in the class of threshold graphs. To attack these kinds of problems we survey several applications of the so-called Kelmans transformation. Mea culpa. This...

Given a group G of automorphisms of a graph Γ, the orbital chromatic polynomial OPΓ,G(x) is the polynomial whose value at a positive integer k is the number of orbits of G on proper k-colorings of Γ. Cameron and Kayibi introduced this polynomial as a means of understanding roots of chromatic polynomials. In this light, they posed a problem asking whether the real roots of the orbital chromatic ...

We study the connections between link invariants, the chromatic polynomial, geometric representations of models of statistical mechanics, and their common underlying algebraic structure. We establish a relation between several algebras and their associated combinatorial and topological quantities. In particular, we define the chromatic algebra, whose Markov trace is the chromatic polynomial χQ ...

1.1 In this note we compute the chromatic polynomial of the Jahangir graph J2p and we prove that it is chromatically unique for p = 3. AMS Subject classification: 05C15

Let P (G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P (H, λ) = P (G, λ) implies H is isomorphic to G. Liu et al. [Liu, R. Y., Zhao, H. X., Ye, C. F.: A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs. Discrete Math., 289, 175–179 (2004)], and Lau and Peng [Lau, G. C., Peng, Y. H.: Chromatic uniquene...

This paper studies various results on chromatic polynomials of graphs. We obtain results on the roots of chromatic polynomials of planar graphs. The main results are chromatic polynomial of a graph is polynomial in integer and the leading coefficient of chromatic polynomial of a graph of order n and size m is one, whose coefficient alternate in sign. Mathematics subject classification 2000: 05C...

We study the connections between link invariants, the chromatic polynomial, geometric representations of models of statistical mechanics, and their common underlying algebraic structure. We establish a relation between several algebras and their associated combinatorial and topological quantities. In particular, we define the chromatic algebra, whose Markov trace is the chromatic polynomial χQ ...

1 The Chromatic Complex 2 1.1 The Chromatic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Quantum Dimension of Graded Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Categorification of the Chromatic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Enhanced States . . . . . . . . . . . . . . . . . . . ....

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...