Motivated by a recent result of Walter [Electron. J. Combin. 16 (2009), R94] concerning the chromatic polynomials of some hypergraphs, we present the chromatic polynomials of several (non-uniform) mixed hypergraphs. We use a recursive process for generating explicit formulae for linear mixed hypercacti and multi-bridge mixed hypergraphs using a decomposition of the underlying hypergraph into bl...

One expansion of the chromatic polynomial n(G,x) of a graph G relies on spanning trees of a graph. In fact, for a connected graph G of order n, one can express n(G,x) = (1 )“-‘x cyi’=;’ ti (1 -x)‘, where ti is the nun&r of spanning trees with external activity 0 and internal activity i. Moreover, it is known (via commutative ring theory) that ti arises as the number of monomials of degree n i 1...

Abstract The excedance set of a permutation π = π1π2 · · ·πk is the set of indices i for which πi > i. We give explicit formulas for the number of permutations whose excedance set is the initial segment {1, 2, . . . ,m} and also of the form {1, 2, . . . ,m,m + 2}. We provide two proofs. The first is an explicit combinatorial argument using rook placements. The second uses the chromatic polynomi...

Abstract. Recently, M. Abért and T. Hubai studied the following problem. The chromatic measure of a finite simple graph is defined to be the uniform distribution on its chromatic roots. Abért and Hubai proved that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. They also showed that the normalized log of the chromatic polynom...

This paper introduces two kinds of graph polynomials, clique polynomial and independent set polynomial. The paper focuses on expansions of these polynomials. Some open problems are mentioned.

2 Preliminaries 2 2.1 Homomorphism numbers and densities . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Local convergence of a graph sequence . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Chromatic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Subtree counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Weight...

Some important properties of the chromatic polynomial also hold for any polynomial set map satisfying pS(x + y) = ∑ T⊎U=S pT (x)pU (y). Using umbral calculus, we give a formula for the expansion of such a set map in terms of any polynomial sequence of binomial type. This leads to several new expansions of the chromatic polynomial. We also describe a set map generalization of Abel polynomials.

There are various ways to define the chromatic polynomial P (G; z) of a graph G. Perhaps the first that springs to mind is to define it to be the graph invariant P (G; k) with the property that when k is a positive integer P (G; k) is the number of colourings of the vertices of G with k or fewer colours such that adjacent vertices receive different colours. One then has to prove that P (G; k) i...