Some remarks on a conjecture of Boyle and Handelman

On a strong form of a conjecture of Boyle and Handelman

Ela on a Strong Form of a Conjecture of Boyle and Handelman

The condition (1.1) on λ1, . . . , λn is a well–known necessary condition for n numbers to be the eigenvalues of an n× n nonnegative matrix (see, for example, Berman and Plemmons [3]). Furthermore, from a result due to Friedland [6, Theorem 1], it is known that (1.1) implies that one of the λi’s is nonnegative and majorizes the moduli of the remaining numbers. Assume for the moment, without los...

Some Remarks on a Conjecture of Alperin

Let G be a finite group, p be a prime, k be an algebraically closed field of characteristic p. J. L. Alperin has conjectured that for any finite group G, # (isomorphism types of simple fcG-modules) = £ # (isomorphism types of projective simple A: (C) where the sum is taken over a set of representatives of the conjugacy classes of /7-subgroups of G. There is also a ' blockwise' version of this c...

An Upper Bound on the Characteristic Polynomial of a Nonnegative Matrix Leading to a Proof of the Boyle–handelman Conjecture

In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if A is an (n+1)×(n+1) nonnegative matrix whose nonzero eigenvalues are: λ0 ≥ |λi|, i = 1, . . . , r, r ≤ n, then for all x ≥ λ0, (∗) r ∏ i=0 (x− λi) ≤ x − λ 0 . To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is tru...

Some remarks on the odd hadwiger's conjecture

We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are two-colored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Gerards and Seymour conjectured that if a graph has no odd complete minor o...