The graph of an abstract polytope

Recently a generalization of simple convex polytopes to combinatorial entities known as abstract polytopes has been proposed. The graph of an abstract polytope of dimensiond is a regular connected graph of degreed. Given a connected regular graph of degreed, it is interesting to find out whether it is the graph of some abstract polytopeP. We obtain necessary and sufficient conditions for this, in terms of the existence of a class of simple cycles in satisfying certain properties. The main result in this paper is that if a pair of simple convex polytopes or abstract polytopes have the same two-dimensional skeleton, then they are isomorphic. Every two-dimensional face of a simple convex polytope or an abstract polytope is a simple cycle in its graph. Given the graph of a simple convex polytope or an abstract polytope and the simple cycles in this graph corresponding to all its two-dimensional faces, then we show how to construct all its remaining faces. Given a regular connected graph and a class of simple cylesD in it, we provide necessary and sufficient conditions under whichD is the class of two-dimensional faces of some abstract polytope which has as its graph.This research has been partially supported by the ISDOS Research Project at the Department of Industrial and Operations Engineering, and by the National Science Foundation under Grant No. GK-27872 with the University of Michigan.     

Segments in enumerating faces

We introduce the concept of a segment of a degenerate convex polytope specified by a system of linear constraints, and explain its importance in developing algorithms for enumerating the faces. Using segments, we describe an algorithm that enumerates all the faces, in time polynomial in their number. The role of segments in the unsolved problem of enumerating the extreme points of a convex polytope specified by a degenerate system of linear constraints, in time polynomial in the number of extreme points, is discussed.Work carried out while on sabbatical leave in the Industrial and Operations Engineering Department at the University of Michigan in Ann Arbor, USA.     

A Benzofuranoid and Two Clerodane Diterpenoids from Pulicaria wightiana

Methyl 5‐formyl‐1‐benzofuran‐6‐carboxylate ( 1 ) and the two clerodane diterpenes, methyl 6‐oxocleroda‐3,13‐dien‐15,16‐olid‐18‐oate ( 2 ) and 2β‐(2‐methylbutanoyl)cleroda‐3,13‐dien‐15,16‐olid‐18‐oic acid ( 3 ), together with 15 known compounds, were isolated from the aerial parts of Pulicaria wightiana. The structures of the new compounds 1, 2 , and 3 were established by spectroscopic (mainly 1D‐ and 2D‐NMR) methods.     

A Study of precipitation morphology in NaCl:Sr and NaCl:Ba crystals

Single crystals of sodium chloride grown from melt containing different concentrations of strontium and barium have been observed to contain a second phase precipitating throughout the interior in the form of thin rods and well defined polygonal platelets. Two types of rods have been observed, one type oriented along 〈100〉 and the other along 〈110〉 directions. The platelets are variously in the form of hexagons, octagons, and rhombs. The precipitates disappear on quenching the crystal and reappear on aging the crystals under suitable conditions.