1. Definitions and a main result 2 1.1. Graphs and colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2. Symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3. Chromatic symmetric functions . . . . . . . . . . . . . . . . . . . . . 5 1.4. Connected components . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5. Circuits and broken circuits . . . . ...

The Ignjatovic theory of chromatic derivatives and series is extended to include other series. In particular series of prolate spheroidal wave functions are used to replace the orthogonal polynomial series in this theory. It is extended further to prolate spheroidal wavelet series that enables us to combine chromatic series with sampling series.

The Bell number B(G) of a simple graph G is the number of partitions of its vertex set whose blocks are independent sets of G. The number of these partitions with k blocks is the (graphical) Stirling number S(G, k) of G. We explore integer sequences of Bell numbers for various one-parameter families of graphs, generalizations of the relation B(Pn) = B(En−1) for path and edgeless graphs, one-par...

Traldi, L., Generalized activities and K-terminal reliability, Discrete Mathematics 96 (1991) 131-149. Suppose each edge of a graph G has a given probability of being useable, and suppose K is some subset of the vertex-set of G. We present a polynomial that is useful in assessing the probability that the elements of K will lie in a particular number of components of the useable portion of G. th...

The identity linking the Tutte polynomial with the Potts model on a graph implies the existence of a decomposition resembling that previously obtained for the chromatic polynomial. Specifically, let {Gn} be a family of bracelets in which the base graph has b vertices. Then the Tutte polynomial of Gn can be written as a sum of terms, one for each partition π of a non-negative integer ` ≤ b: (x− ...

The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion-contraction relations, its specialisation to cellularly embedded graphs does not. Here we ex...

We study the chromatic polynomial PG(q) for m× n squareand triangular-lattice strips of widths 2 ≤ m ≤ 8 with cyclic boundary conditions. This polynomial gives the zero-temperature limit of the partition function for the antiferromagnetic q-state Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin–Kasteleyn representation for such lattices and obtai...

Motivated by the study of Macdonald polynomials, J. Haglund and A. Wilson introduced a nonsymmetric polynomial analogue chromatic quasisymmetric function called chromatic polynomial Dyck graph. We give positive expansion for this in basis fundamental slide polynomials using recent work Assaf-Bergeron on flagged <mml:math xmlns:mml="ht...

The chromatic polynomialχG(q) of a graph G counts the num-ber of proper colorings of G. We give an affirmative answer to the conjectureof Read and Rota-Heron-Welsh that the absolute values of the coefficients ofthe chromatic polynomial form a log-concave sequence. The proof is obtainedby identifyingχG(q) with a sequence of numerical invariants of a projectivehypersur...

We show that one can compute the injective chromatic number of a chordal graph G at least as efficiently as one can compute the chromatic number of (G−B), where B are the bridges of G. In particular, it follows that for strongly chordal graphs and so-called power chordal graphs the injective chromatic number can be determined in polynomial time. Moreover, for chordal graphs in general, we show ...