‎we investigate two constructions‎ - ‎the replacement and the zig-zag‎ ‎product of graphs‎ - ‎describing several fascinating connections‎ ‎with combinatorics‎, ‎via the notion of expander graph‎, ‎group‎ ‎theory‎, ‎via the notion of semidirect product and cayley graph‎, ‎and‎ ‎with markov chains‎, ‎via the lamplighter random walk‎. ‎many examples‎ ‎are provided‎.

It is known that the expansion property of a graph influences the performance of the corresponding code when decoded using iterative algorithms. Certain graph products may be used to obtain larger expander graphs from smaller ones. In particular, the zig-zag product and replacement product may be used to construct infinite families of constant degree expander graphs. This paper investigates the...

In this paper, Kekule structures of benzenoid chains are considered. It has been shown that the coefficients of a B_n (x) Morgan-Voyce polynomial equal to the number of k-matchings (m(G,k)) of a path graph which has N=2n+1 points. Furtermore, two relations are obtained between regularly zig-zag nonbranched catacondensed benzenid chains and Morgan-Voyce polynomials and between regularly zig-zag ...

Vertebrate and invertebrate genomes contain scores of small secreted or transmembrane proteins with two immunoglobulin (Ig) domains. Many of them are expressed in the nervous system, yet their function is not well understood. We analyze here knockout alleles of all eight members of a family of small secreted or transmembrane Ig domain proteins, encoded by the Caenorhabditis elegans zig ("zwei I...

During nervous system development, neuronal cell bodies and their axodendritic projections are precisely positioned through transiently expressed patterning cues. We show here that two neuronally expressed, secreted immunoglobulin (Ig) domain-containing proteins, ZIG-5 and ZIG-8, have no detectable role during embryonic nervous system development of the nematode Caenorhabditis elegans but are j...

Vertebrate and invertebrate genomes contain scores of small secreted or transmembrane proteins with two immunoglobulin (Ig) domains. Many of them are expressed in the nervous system, yet their function is not well understood. We analyze here knockout alleles of all 8 members of a family of small secreted or transmembrane Ig domain proteins, encoded by the C. elegans zig ("zwei Ig Domänen") gene...

an experiment was conducted at zahak research station in zabol in 1996-97 to determine the best method of planting and plant density for mixed culture of cucumber and eggplant. in this experiment, yields of two crops were compared in individual and mixed cultures. in both systems the distances between two plant were 25 and 50 cm, and in the mixed culture eggplants were planted in the opposite o...