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...

The expander constructions based on algebraic methods can give expanders that are both explicit (i.e. we can quickly construct the graph, or even obtain neighborhood information without constructing the entire graph, and Ramanujan, meaning that the spectral gap is essentially as large as possible. It also follows from this spectral bound that the edge expansion of Ramanujan graphs is essentiall...

‎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.

Reingold et al. introduced the notion zig-zag product on two different graphs, and presented a fully explicit construction of dregular expanders with the second largest eigenvalue O(d−1/3). In the same paper, they ask whether or not the similar technique can be used to construct expanders with the second largest eigenvalue O(d−1/2). Such graphs are called Ramanujan graphs. Recently, zig-zag pro...

The second Zagreb coindex is a well-known graph invariant defined as the total degree product of all non-adjacent vertex pairs in a graph. The second Zagreb eccentricity coindex is defined analogously to the second Zagreb coindex by replacing the vertex degrees with the vertex eccentricities. In this paper, we present exact expressions or sharp lower bounds for the second Zagreb eccentricity co...

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, startin...

In this paper, we study the problem of testing the conductance of a given graph in the general graph model. Given distance parameter ε and any constant σ > 0, there exists a tester whose running time isO( (1+σ)/2·logn·log ε ε·Φ2 ), where n is the number of vertices andm is the number of edges of the input graph. With probability at least 2/3, the tester accepts all graphs of conductance at leas...

Zinc alpha-2-glycoprotein (ZAG) is a M(r) 41,000 glycoprotein secreted by a variety of normal epithelia. ZAG was recently shown to stimulate lipolysis in adipocytes, leading to the development of cachexia in animals with ZAG-producing tumors. To understand the possible contribution of ZAG to the development of cachexia in men with prostate cancer, ZAG production by normal and malignant prostate...

The generalized hierarchical product of graphs was introduced by L. Barriére et al in 2009. In this paper, reformulated first Zagreb index of generalized hierarchical product of two connected graphs and hence as a special case cluster product of graphs are obtained. Finally using the derived results the reformulated first Zagreb index of some chemically important graphs such as square comb latt...