Intersection Matrices Revisited

Several intersection matrices of s‐subsets versus k‐subsets of a v‐set are introduced in the literature. We study these matrices systematically through counting arguments and generating function techniques. A number of new or known identities appear as natural consequences of this viewpoint; especially, use of the derivative operator and some related operators reveals some connections between intersection matrices and the “combinatorics of creation‐annihilation.” As application, the eigenvalues of several intersection matrices including some generalizations of the adjacency matrices of the Johnson scheme are derived; two new bases for the Bose–Mesner algebra of the Johnson scheme are introduced and the associated intersection numbers are obtained as well. Finally, we determine the rank of some intersection matrices. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 383–397, 2012     

Iron Perchlorate on Silica Gel as Multi-purpose Reagent for Catalysis of Closure and Rupture of Carbon-Oxygen Bond in Epoxides,Alcohols, and Esters

Aliphatic alcohols and water in the presence of catalytic amounts of Fe³⁺ ion introduced as acids esterification with various type alcohols was carried out using the system Fe(ClO₄)₃-silica gel in dichloromethane under conditions excluding solvolysis. Acetylation and formylation of alcohols was performed by efficient transesterification with ethyl acetate and ethyl formate.     

On zero-sum -flows of graphs

For a graph G, a zero-sum flow is an assignment of non-zero real numbers on the edges of G such that the total sum of all edges incident with any vertex of G is zero. A zero-sum k-flow for a graph G is a zero-sum flow with labels from the set {±1,…,±(k-1)}. In this paper for a graph G, a necessary and sufficient condition for the existence of zero-sum flow is given. We conjecture that if G is a graph with a zero-sum flow, then G has a zero-sum 6-flow. It is shown that the conjecture is true for 2-edge connected bipartite graphs, and every r-regular graph with r even, r>2, or r=3.     

A Facile One‐Pot Synthesis of Functionalized 1,5‐Dihydro‐2H‐[1]benzopyrano[2,3‐b]pyridin‐5‐ones

The highly reactive 1 : 1 intermediate generated in the reaction between dialkyl acetylenedicarboxylate (=but‐2‐ynedioic acid dialkyl ester) 4 and triphenylphosphine was trapped by 2‐amino‐4‐oxo‐4H‐1‐benzopyran‐3‐carboxaldehydes 5 to yield highly functionalized dialkyl‐1,5‐dihydro‐5‐oxo‐1‐phenyl‐2H‐[1]benzopyrano[2,3‐b]pyridine‐2,3‐dicarboxylates in high yield.