Zig-zag Graph Research Articles

On Tades of Disjoint Union of Some Graphs

Consider a total labeling ξ of a graph G . For every two different edges e and f of G , let w t ( e ) ≠ w t ( f ) where weight of e = x y is defined as w t ( e ) = | ξ ( e ) – ξ ( x ) – ξ ( y ) | . Then ξ is called edge irregular total absolute difference k -labeling of G . Let k be the minimum integer for which there is a graph G with edge irregular total absolute difference labeling. This k is called the total absolute difference edge irregularity strength of the graph G , denoted t a d e s ( G ) . We compute t a d e s of S C n , disjoint union of grid and zigzag graph.

Semifinite Harmonic Functions on the Zigzag Graph

We study semifinite harmonic functions on the zigzag graph, which corresponds to the Pieri rule for the fundamental quasisymmetric functions $$\{F_{\lambda}\}$$ . The main problem, which we solve here, is to classify the indecomposable semifinite harmonic functions on this graph. We show that these functions are in a natural bijective correspondence with some combinatorial data, the so-called semifinite zigzag growth models. Furthermore, we describe an explicit construction that produces a semifinite indecomposable harmonic function from every semifinite zigzag growth model. We also establish a semifinite analogue of the Vershik–Kerov ring theorem.

Compositions with restricted parts

Euler showed that the number of partitions of n into distinct parts equals the number of partitions of n into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on partitions with restricted parts, which were confirmed analytically by Andrews and Chern and combinatorially by Yang.Analogous to Euler’s partition theorem, it is known that the number of compositions of n with odd parts equals the number of compositions of n+1 with parts greater than one, as both numbers equal the Fibonacci number Fn. Recently, Sills provided a bijective proof for this result using binary sequences, and Munagi proved a generalization similar to Glaisher’s result using the zigzag graphs of compositions. Extending Sills’ bijection, we obtain a further generalization which is analogous to Franklin’s result. We establish, both analytically and combinatorially, two closed formulae for the number of compositions with restricted parts appearing in our generalization. We also prove some composition analogues for the conjectures of Beck.

Pebbling on zig-zag chain graph of n odd cycles

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The pebbling number of G, f (G), is the least n such that any distribution of n pebbles on G allows one pebble to be reached to any specified, but an arbitrary vertex. Similarly, the t−pebbling number of G, ft(G), is the least m such that from any distribution of m pebbles, we can move t pebbles to any specified, but an arbitrary vertex. In this paper, we determine the pebbling number, and the t−pebbling number of the zigzag chain graph of n copies of odd cycles.

On the edge irregularity strength of grid graphs

For a simple graph G, a vertex labeling ϕ:V(G)→{1,2,…,k} is called a vertex k-labeling. For any edge xy in G, its weight wϕ(xy)=ϕ(x)+ϕ(y). If all the edge weights are distinct, then ϕ is called an edge irregular k-labeling of G. The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G).In this paper, we determine an exact value of edge irregularity strength for triangular grid graph Lnm, zigzag graph Znm and Cartesian product Pn□Pm□P2.

Zig-Zag Graphs and Partition Identities of A. K. Agarwal

We present new proofs of certain partition identities which appeared in three published articles of Ashok K. Agarwal. The main tool is obtained by combining zig-zag graphs with their geometric connection to partition Ferrers graphs as hinted by MacMahon in his classic text Combinatory Analysis. We give combinatorial proofs of the identities and provide a generalization or refinement of each result.

Single-valued multiple polylogarithms and a proof of the zig–zag conjecture

A long-standing conjecture in quantum field theory due to Broadhurst and Kreimer states that the periods of the zig–zag graphs are a certain explicit rational multiple of the odd values of the Riemann zeta function. In this paper we prove this conjecture by constructing a certain family of single-valued multiple polylogarithms which correspond to multiple zeta values ζ(2,…,2,3,2,…2) and using the method of graphical functions. The zig–zag graphs are the only infinite family of primitive graphs in ϕ44 theory (in fact, in any renormalisable quantum field theory in four dimensions) whose periods are now known.

Randomness-Efficient Sampling within NC1

We construct a randomness-efficient averaging sampler that is computable by uniform constant-depth circuits with parity gates (i.e., in uniform AC 0[⊕]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC 1. For example, we obtain the following results: Our sampler is based on the zig-zag graph product of Reingold, Vadhan & Wigderson (Annals of Math 2002) and as part of our analysis we givean elementary proof of a generalization of Gillman’s Chernoff Bound for Expander Walks (SIAM Journal on Computing 1998).

Zig-zag and replacement product graphs and LDPC codes

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 use of zig-zag and replacement product graphs for the construction of codes on graphs. A modification of the zig-zag product is also introduced, which can operate on two unbalanced biregular bipartite graphs, and a proof of the expansion property of this modified zig-zag product is presented.

Concatenated zigzag hadamard codes

In this correspondence, we introduce a new class of low-rate error correction codes called zigzag Hadamard (ZH) codes and their concatenation schemes. Each member of this class of codes is specified by a highly structured zigzag graph with each segment being a Hadamard codeword. The ZH codes enjoy extremely simple encoding and very low-complexity soft-input-soft-output (SISO) decoding based on a posteriori probability (APP) fast Hadamard transform (FHT) technique. We present an asymptotic performance analysis of the proposed concatenated ZH codes using the extrinsic mutual information transfer (EXIT) chart for infinite-length codes. We also provide a union bound analysis of the error performance for finite-length codes. Furthermore, the concatenated ZH codes are shown to be a good class of codes in the low-rate region. Specifically, a rate-0.0107 concatenated code with three ZH components and an interleaver size of 65536 can achieve the bit error rate (BER) performance of 10/sup -5/ at -1.15dB, which is only 0.44 dB away from the ultimate Shannon limit. The proposed concatenated ZH codes offer similar performance as another class of low-rate codes-the turbo-Hadamard codes, and better performance than superorthogonal turbo codes, with much lower encoding and decoding complexities.

New near shannon limit codes: Zigzag codes and concatenated zigzag codes

This paper introduces a family of error-correcting codes called zigzag codes. A zigzag code is described by a highly structured zigzag graph. Due to the structural properties of the graph, very low-complexity soft-in, soft-out decoding rules can be implemented. We present a decoding rule, based on the Max-Log-APP(MLA) formulation, which requires a total of only 20 addition-equivalent-operations per information bit per iteration. Simulation of a rate-1/2 concatenated zigzag code with four constituent encoders with interleaver length 65536 yields a bit error rate (BER) and of 10−5 at 0.9 dB and 1.4 dB away from the Shannon limit by optimal (APP) and low-cost sub-optimal (MLA) decoders, respectively.

Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders

The main contribution of this work is a new type of graph product, which we call the zig-zag product. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! Iteration yields simple explicit constructions of constant-degree expanders of arbitrary size, starting from one constant-size expander. Crucial to our intuition (and simple analysis) of the properties of this graph product is the view of expanders as functions which act as wave propagators -they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated. In these terms, the graph product affords the constructive interference of two such waves.

Zigzag codes and concatenated zigzag codes

This paper introduces a family of error-correcting codes called zigzag codes. A zigzag code is described by a highly structured zigzag graph. Due to the structural properties of the graph, very low-complexity soft-in/soft-out decoding rules can be implemented. We present a decoding rule, based on the Max-Log-APP (MLA) formulation, which requires a total of only 20 addition-equivalent operations per information bit, per iteration. Simulation of a rate-1/2 concatenated zigzag code with four constituent encoders with interleaver length 65 536, yields a bit error rate (BER) of 10/sup -5/ at 0.9 dB and 1.3 dB away from the Shannon limit by optimal (APP) and low-cost suboptimal (MLA) decoders, respectively. A union bound analysis of the bit error probability of the zigzag code is presented. It is shown that the union bounds for these codes can be generated very efficiently. It is also illustrated that, for a fixed interleaver size, the concatenated code has increased code potential as the number of constituent encoders increases. Finally, the analysis shows that zigzag codes with four or more constituent encoders have lower error floors than comparable turbo codes with two constituent encoders.

Cohomology of graph hypersurfaces associated to certain Feynman graphs

To any Feynman graph (with 2n edges) we can associate a hypersurface $X\subset\PP^{2n-1}$. We study the middle cohomology $H^{2n-2}(X)$ of such hypersurfaces. S. Bloch, H. Esnault, and D. Kreimer (Commun. Math. Phys. 267, 2006) have computed this cohomology for the first series of examples, the wheel with spokes graphs $WS_n$, $n\geq 3$. Using the same technique, we introduce the generalized zigzag graphs and prove that $W_5(H^{2n-2}(X))=\QQ(-2)$ for all of them (with $W_{*}$ the weight filtration). Next, we study primitively log divergent graphs with small number of edges and the behavior of graph hypersurfaces under the gluing of graphs.