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In graph theory, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs G 1 and G 2 and produces a graph H with the following properties: The vertex set of H is the Cartesian product V ( G 1 ) × V ( G 2 ) , where V ( G 1 ) and V ( G 2 ) are the vertex sets of G 1 and G 2 , respectively.
The expansion of a graph can be measured by its spectral gap, with an important property of the zigzag product the preservation of the spectral gap.That is, if is a “good enough” expander (has a large spectral gap) then the expansion of the zigzag product is close to the original expansion of .
The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i The graph of y = W(x) for real x < 6 and y > −4. The upper branch (blue) with y ≥ −1 is the graph of the function W 0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W −1. The minimum value of x is ...
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis.
MATLAB (an abbreviation of "MATrix LABoratory" [22]) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks.MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages.
Thus, the Cartesian product of two hypercube graphs is another hypercube: Q i Q j = Q i+j. The Cartesian product of two median graphs is another median graph. The graph of vertices and edges of an n-prism is the Cartesian product graph K 2 C n. The rook's graph is the Cartesian product of two complete graphs.
The 'bathtub curve' hazard function (blue, upper solid line) is a combination of a decreasing hazard of early failure (red dotted line) and an increasing hazard of wear-out failure (yellow dotted line), plus some constant hazard of random failure (green, lower solid line). The bathtub curve is a particular shape of a failure rate graph.
Kronecker graphs are a construction for generating graphs for modeling systems. The method constructs a sequence of graphs from a small base graph by iterating the Kronecker product. [1] A variety of generalizations of Kronecker graphs exist. [2] The Graph500 benchmark for supercomputers is based on the use of a stochastic version of Kronecker ...