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.

For a given graph G, let P (G,λ) be the chromatic polynomial of G, where λ is considered to be a real number. In this paper, we study the bounds for P (G,λ)/P (G,λ − 1) and P (G,λ)/P (G − x, λ), where x is a vertex in G, λ ≥ n and n is the number of vertices of G.

In many games, players’ decisions consist of multiple subdecisions, and hence can give rise to an exponential number of pure strategies. However, this set of pure strategies is often structured, allowing it to be represented compactly, as in network congestion games, security games, and extensive form games. Reduction to the standard normal form generally introduces exponential blow-up in the s...

Boolean functions (BFs) are central in many fields of engineering and mathematics, such as cryptography, circuit design, and combinatorics. Moreover, they provide a simple framework for studying neural computation mechanisms of the brain. Many representation schemes for BFs exist to satisfy the needs of the domain they are used in. In neural computation, it is of interest to know how many input...

The projective hull X̂ of a compact set X ⊂ P is an analogue of the classical polynomial hull of a set in C. In the special case that X ⊂ C ⊂ P, the affine part X̂ ∩C can be defined as the set of points x ∈ C for which there exists a constant Mx so that |p(x)| ≤ M x sup X |p| for all polynomials p of degree ≤ d, and any d ≥ 1. Let X̂(M) be the set of points x where Mx can be chosen ≤ M . Using an ...

We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for cer...

We present a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. Examples of two knots with the same Alexander module but different twisted Alexander polynomials are given.

We show that the Lawrence–Krammer representation is unitary. We explicitly present the non-singular matrix representing the sesquilinear pairing invariant under the action. We show that reversing the orientation of a braid is equivalent to the transposition of its Lawrence–Krammer matrix followed by a certain conjugation. As corollaries it is shown that the characteristic polynomial of the Lawr...

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...