ON POLYTOPAL UPPER BOUND SPHERES

On Polytopal Upper Bound Spheres

Generalizing a result (the case k = 1) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension 2k+1 belongs to the generalizedWalkup class Kk(2k + 1), i.e., all its vertex links are k-stacked spheres. This is surprising since the k-stacked spheres minimize the face-vector (among all polytopal spheres with given f0, . . . , fk−1) while the upper bound spheres maximize...

An upper bound on the volume of discrete spheres

The number M,(g) of nonzero elements a in F is M2(g) = 22'-1-Ml(g). Now using IS(= w[ p,(x)] = w , ns = 2,'-1, 0 Corollary 4.7: If a) F,k is the splitting field of x "-1 over F,, b) If k = 2t, s = 2d + 1, where d is a divisor of t , and if g(x) is a primitiue divisor of x s-1 over F,, then the weight distribution of C is given by Theorem 4.5 for all integer w such that w is even and and Proposi...

Kalai's squeezed 3-spheres are polytopal

Enumeration in Convex Geometries and Associated Polytopal Subdivisions of Spheres

We construct CW spheres from the lattices that arise as the closed sets of a convex closure, the meet-distributive lattices. These spheres are nearly polytopal, in the sense that their barycentric subdivisions are simplicial polytopes. The complete information on the numbers of faces and chains of faces in these spheres can be obtained from the defining lattices in a manner analogous to the rel...

An Upper Bound on the First Zagreb Index in Trees

In this paper we give sharp upper bounds on the Zagreb indices and characterize all trees achieving equality in these bounds. Also, we give lower bound on first Zagreb coindex of trees.