Given a topological space T and a finite subset T0 of T , we associate two graphs with T and T0 that, under rather mild conditions, turn out to be a block graph and a tree, respectively. This construction is of interest, e.g., in the context of phylogenetic analysis where T may model a full ‘‘orbit’’ of a dynamical branching process, and T0 the set of its branching points. © 2008 Elsevier Ltd. ...

Classical invariant theory has died and been resurrected many times. Its golden age in the last century was marked with the flowering of unsurpassed computational ability, and the explicit determination of the invariants of most of the elementary polynomials. The computational approach is commonly acknowledged to have been dealt a death-blow by Hilbert’s celebrated Basis Theorem, an existential...

A surprising result of FitzGerald and Horn (1977) shows that A◦α := (aα ij) is positive semidefinite (p.s.d.) for every entrywise nonnegative n× n p.s.d. matrix A = (aij) if and only if α is a positive integer or α ≥ n− 2. Given a graph G, we consider the refined problem of characterizing the setHG of entrywise powers preserving positivity for matrices with a zero pattern encoded by G. Using al...

It is conjectured that the Khovanov homology of a knot is invariant under mutation. In this paper, we review the spanning tree complex for Khovanov homology, and reformulate this conjecture using a matroid obtained from the Tait graph (checkerboard graph) G of a knot diagram K. The spanning trees of G provide a filtration and a spectral sequence that converges to the reduced Khovanov homology o...

in this paper we introduce the concept of generalized baer-invariant of a pair of groups with respect to two varieties ? and ? of groups. we give some inequalities for the generalized baer-invariant of a pair of finite groups, when ? is considered to be the schur-baer variety. further, we present a sufficient condition under which the order of the generalized baer-invariant of a pair of finite ...

The product P◦Q of graph properties P,Q is a class of all graphs having a vertex-partition into two parts inducing subgraphs with properties P and Q, respectively. For a graph invariant φ and a graph property P we define φ(P) as the minimum of φ(F ) taken over all minimal forbidden subgraphs F of P . An invariant of graph properties φ is said to be additive with respect to reducible hereditary ...

Given a weighted graph Gx, where (x(v) : v ∈ V ) is a non-negative, realvalued weight assigned to the vertices of G, let B(Gx) be an upper bound on the fractional chromatic number of the weighted graph Gx; so χf (Gx) ≤ B(Gx). To investigate the worst-case performance of the upper bound B, we study the graph invariant β(G) = sup x 6=0 B(Gx) χf (Gx) . In recent work a particular upper bound resul...

In the present paper we determine the Thurston-Bennequin invariant of graph divide links, which include all closed positive braids, all divide links and certain negative twist knots. As a corollary of this and a result of P. Lisca and A.I. Stipsicz, we prove that the 3-manifold obtained from S by Dehn surgery along a non-trivial graph divide knot K with coefficient r carries positive, tight con...

We use the graph transform method to prove existence of invariant manifolds near fixed points of maps tangent to invariant subspaces of the linearization. In contrast to the best known of such theorems, we do not assume that the corresponding space for the linear map is a spectral subspace. Indeed, we allow that there is no invariant complement (in particular, we do not need that the decomposit...