In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number and the harmonious chromatic number of P4-tidy graphs and (q, q − 4)-graphs, for every fixed q. These classes include cographs, P4-sparse and P4-lite graphs. We also obtain a polynomial time algorithm to determine the Grundy number of (q, q − 4)-graphs. All these coloring pro...

A number of mathematical relations for the sextet polynomial are derived. A graph has been introduced (the so called C-graph), representing those properties of a benzenoid system which are essential in the sextet theory of Clar. The main structural properties of the C-graph are determined. The obtained results contribute towards a better understanding of the algebraic and combinatorial backgrou...

We introduce a domination polynomial of a graph G. The domination polynomial of a graph G of order n is the polynomial D(G, x) = ∑n i=γ(G) d(G, i)x , where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. We obtain some properties of D(G, x) and its coefficients. Also we compute this polynomial for some specific graphs.

In this paper we present several results and open problems about logconcavity properties of sequences associated with graph colorings. Five polynomials intimately related to the chromatic polynomial of a graph are introduced and their zeros, combinatorial and log-concavity properties are studied. Four of these polynomials have never been considered before in the literature and some yield new ex...

Two unitary integral transforms with a very-well poised 7F6-function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The 7F6-function involved can be considered as a non-polynomial extension of the Wilson polynomial, and is therefore called a Wilson function. The two integral transforms are called a Wil...

This paper describes an improvement in the upper bound for the magnitude of a coefficient of a term in the chromatic polynomial of a general graph. If ar is the coefficient of the q r term in the chromatic polynomial P (G, q), where q is the number of colors, then we find ar ≤ ( e v−r ) − ( e−g+2 v−r−g+2 ) + ( e−kg−g+2 v−r−g+2 ) − ∑kg−lg n=1 ∑lg−1 m=1 ( e−g+1−n−m v−r−g ) − δg,3 ∑kg+l ∗ g+1 −lg ...

Let G = (V,E) be a 2-connected plane graph on n vertices with outer face C such that every 2-vertex cut of G contains at least one vertex of C. Let PG(q) denote the chromatic polynomial of G. We show that (−1)nPG(q) > 0 for all 1 < q ≤ 1.2040.... This result is a corollary of a more general result that (−1)ZG(q,w) > 0 for all 1 < q ≤ 1.2040..., where ZG(q,w) is the multivariate Tutte polynomial...

We establish closed-form expansions for the universal edge elimination polynomial of paths and cycles and their generating functions. This includes closed-form expansions for the bivariate matching polynomial, the bivariate chromatic polynomial, and the covered components polynomial.

The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these...

We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce the class of (hyper)graph polynomials definable in second order logic, and outline a research progr...