Concerning a conjecture of R. L. Graham

Partial proof of Graham Higman's conjecture related to coset diagrams

Graham Higman has defined coset diagrams for PSL(2,ℤ). These diagrams are composed of fragments, and the fragments are further composed of two or more circuits. Q. Mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

On a conjecture of Butler and Graham

In this paper we prove a conjecture of Bulter and Graham [2] on the existence of a certain way of marking the lines in [k] for any prime k. The conjecture states that there exists a way of marking each line of [k] one point so that every point in [k] is marked exactly a or b times as long as the parameters (a, b, n, k) satisfy the condition that there are integers s, t such that s + t = k and a...

Note on a conjecture of Graham

An old conjecture of Graham stated that if [Formula: see text] is a prime and [Formula: see text] is a sequence of [Formula: see text] terms from the cyclic group [Formula: see text] such that all (nontrivial) zero-sum subsequences have the same length, then [Formula: see text] must contain at most two distinct terms. In 1976, Erdős and Szemerédi gave a proof of the conjecture for sufficiently ...

CONCERNING INTERSECTING CONTINUA BY R. L. MooRu

Proof of a conjecture of Graham and Lovász concerning unimodality of coefficients of the distance characteristic polynomial of a tree

We establish a conjecture of Graham and Lovász that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal; we also prove they are log-concave. Email addresses: [email protected] (Ghodratollah Aalipour), [email protected] (Aida Abiad), [email protected] (Zhanar Berikkyzy), [email protected] (Leslie Hogben), [email protected] (Fra...