Recently, Stanley [21] has defined a symmetric function generalization of the chromatic polynomial of a graph. Independently, Chung and Graham [4] have defined a digraph polynomial called the cover polynomial which is closely related to the chromatic polynomial of a graph (in fact, as we shall see, the cover polynomial of a certain digraph associated to a poset P coincides with the chromatic po...

Chromatic polynomials are important objects in graph theory and statistical physics, but as a result of computational difficulties, their study is limited to graphs that are small, highly structured, or very sparse. We have devised and implemented two algorithms that approximate the coefficients of the chromatic polynomial P (G,x), where P (G, k) is the number of proper k-colorings of a graph G...

It is proved that the chromatic polynomial of a connected graph with n vertices and m edges has a root with modulus at least (m&1) (n&2); this bound is best possible for trees and 2-trees (only). It is also proved that the chromatic polynomial of a graph with few triangles that is not a forest has a nonreal root and that there is a graph with n vertices whose chromatic polynomial has a root wit...

We consider proper colorings of planar graphs embedded in the annulus, such that vertices on one rim can take Qs colors, while all remaining vertices can take Q colors. The corresponding chromatic polynomial is related to the partition function of a boundary loop model. Using results for the latter, the phase diagram of the coloring problem (with real Q and Qs) is inferred, in the limits of two...

It is shown how to compute the Chromatic Polynomial of a simple graph utilizing bond lattices and the Möbius Inversion Theorem, which requires the establishment of a refinement ordering on the bond lattice and an exploration of the Incidence Algebra on a partially ordered set.

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)...

Let P (k) be the chromatic polynomial of a graph with n ≥ 2 vertices and no isolated vertices, and let R(k) = P (k + 1)/k(k + 1). We show that the coefficients of the polynomial (1 − t) ∑ ∞ k=1 R(k)t are nonnegative and we give a combinatorial interpretation to R(k) when k is a nonpositive integer.

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.

The injective chromatic number χi(G) [5] of a graph G is the minimum number of colors needed to color the vertices of G such that two vertices with a common neighbor are assigned distinct colors. In this paper we define injective chromatic sum and injective strength of a graph and obtain the injective chromatic sum of complete graph, paths, cycles, wheel graph and complete bipartite graph. We a...