For any positive integers a, b, c, d, let Ra,b,c,d be the graph obtained from the complete graphs Ka,Kb,Kc and Kd by adding edges joining every vertex in Ka and Kc to every vertex in Kb and Kd. This paper shows that for arbitrary positive integers a, b, c and d, every root of the chromatic polynomial of Ra,b,c,d is either a real number or a non-real number with its real part equal to (a + b + c...

In this paper, using a standard method of computing the chromatic polynomial of hypergraphs, we introduce a new reduction theorem which allows us to find explicit formulae for the chromatic polynomials of some (complete) non-uniform (m, l)− hyperwheels and non-uniform (m, l)−hyperfans. These hypergraphs, constructed through a “join” graph operation, are some generalizations of the well-known wh...

The notion of activities with respect to spanning trees in graphs was introduced by W.T. Tutte, and generalized to activities with respect to bases in matroids by H. Crapo. We present a further generalization, to activities with respect to arbitrary subsets of matroids. These generalized activities provide a unified view of several different expansions of the Tutte polynomial and the chromatic ...

The chromatic symmetric function XG of a graph G was introduced by Stanley. In this paper we introduce a quasisymmetric generalization X G called the k-chromatic quasisymmetric function of G and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of XG to χG(λ), the chromatic polynomial, we also define a generalization χ k G(λ) and sh...

One of the remarkable features of the chromatic polynomial χ(Q) is Tutte’s golden identity. This relates χ(φ+ 2) for any triangulation of the sphere to (χ(φ+ 1)) for the same graph, where φ denotes the golden ratio. We show that this result fits in the framework of quantum topology and give a proof of Tutte’s identity using the notion of the chromatic algebra, whose Markov trace is the chromati...

We define a class of bipartite graphs that correspond naturally with Ferrers diagrams. We give expressions for the number of spanning trees, the number of Hamiltonian paths when applicable, the chromatic polynomial and the chromatic symmetric function. We show that the linear coefficient of the chromatic polynomial is given by the excedance set statistic.

We discuss the complexity of computing various graph polynomials of graphs of fixed clique-width. We show that the chromatic polynomial, the matching polynomial and the two-variable interlace polynomial of a graph G of clique-width at most k with n vertices can be computed in time O(n), where f(k) ≤ 3 for the inerlace polynomial, f(k) ≤ 2k + 1 for the matching polynomial and f(k) ≤ 3 · 2 for th...

We obtain new necessary conditions on a graph which shares the same chromatic polynomial as that of the complete tripartite graph Km,n,r. Using these, we establish the chromatic equivalence classes for K1,n,n+1 (where n ≥ 2). This gives a partial solution to a question raised earlier by the authors. With the same technique, we further show that Kn−3,n,n+1 is chromatically unique if n ≥ 5. In th...

Given a finite simple graphG = (V,E) with chromatic number c and chromatic polynomial C(x). Every vertex graph coloring f of G defines an index if (x) satisfying the Poincaré-Hopf theorem [17] ∑ x if (x) = χ(G). As a variant to the index expectation result [19] we prove that E[if (x)] is equal to curvature K(x) satisfying Gauss-Bonnet ∑ xK(x) = χ(G) [16], where the expectation is the average ov...

Let P (G,λ) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, written G ∼ H, if P (G,λ) = P (H,λ). A graph G is chromatically unique, written χ−unique, if for any graph H, G ∼ H implies that G is isomorphic with H. In this paper we prove the chromatic uniqueness of a new family of 6-bridge graphs.