Article history: Received 6 August 2015 Received in revised form 28 January 2016 Accepted 29 January 2016 Available online 11 February 2016

For a simplicial complex ∆ we study the effect of barycentric subdivision on ring theoretic invariants of its StanleyReisner ring. In particular, for Stanley-Reisner rings of barycentric subdivisions we verify a conjecture by Huneke and Herzog & Srinivasan, that relates the multiplicity of a standard graded k-algebra to the product of the maximal shifts in its minimal free resolution up to the ...

Consider the barycentric subdivision which cuts a given triangle along its medians to produce six new triangles. Uniformly choosing one of them and iterating this procedure gives rise to a Markov chain. We show that almost surely, the triangles forming this chain become flatter and flatter in the sense that their isoperimetric values goes to infinity with time. Nevertheless, if the triangles ar...

Article history: Received 4 November 2015 Received in revised form 21 April 2016 Accepted 23 April 2016 Available online xxxx

The notion of homogeneous barycentric coordinates provides a powerful tool of analysing problems in plane geometry. In this paper, we explain the advantages over the traditional use of trilinear coordinates, and illustrate its powerfulness in leading to discoveries of new and interesting collinearity relations of points associated with a triangle.

This paper proposes a simple and efficient technique to generate uniformly random points in a tetrahedron. The technique generates random points in a cube, and folds the cube into the barycentric space of the tetrahedron in a way that preserves uniformity.

Gross and Rosen asked if the genus of a 2-dimensional complex K embeddable in some (orientable) surface is equal to the genus of the graph of appropriate barycentric subdivision of K. We answer the nonorientable genus and the Euler genus versions of Gross and Rosen's question in affirmative. We show that this is not the case for the orientable genus by proving that taking blog2 gcth barycentric...

We show that if a three-dimensional polytopal complex has a knot in its 1-skeleton, where the bridge index of the knot is larger than the number of edges of the knot, then the complex is not constructible, and hence, not shellable. As an application we settle a conjecture of Hetyei concerning the shellability of cubical barycentric subdivisions of 3-spheres. We also obtain similar bounds conclu...

The spectrum of the Laplacian of successive Barycentric subdivisions of a graph converges exponentially fast to a limit which only depends on the clique number of the initial graph and not on the graph itself. Announced in [40]), the proof uses now an explicit linear operator mapping the clique vector of a graph to the clique vector of the Barycentric refinement. The eigenvectors of its transpo...