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Circulant graphs can be described in several equivalent ways: [2] The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph's vertices. In other words, the graph has an automorphism which is a cyclic permutation of its vertices. The graph has an adjacency matrix that is a circulant matrix.
For example, in 2-space n = 2, a rotation by angle θ has eigenvalues λ = e iθ and λ = e −iθ, so there is no axis of rotation except when θ = 0, the case of the null rotation. In 3-space n = 3, the axis of a non-null proper rotation is always a unique line, and a rotation around this axis by angle θ has eigenvalues λ = 1, e iθ, e −iθ.
In graph theory, a graph or digraph whose adjacency matrix is circulant is called a circulant graph/digraph. Equivalently, a graph is circulant if its automorphism group contains a full-length cycle. The Möbius ladders are examples of circulant graphs, as are the Paley graphs for fields of prime order.
A circle with five chords and the corresponding circle graph. In graph theory, a circle graph is the intersection graph of a chord diagram.That is, it is an undirected graph whose vertices can be associated with a finite system of chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other.
For example, a submatrix taken from rows 2 through 4 and columns 3 through 4 can be written as: >> A ( 2 : 4 , 3 : 4 ) ans = 11 8 7 12 14 1 A square identity matrix of size n can be generated using the function eye , and matrices of any size with zeros or ones can be generated with the functions zeros and ones , respectively.
With an additional (free) package, it's also possible to generate SVG-graphs with R directly. See an example with code on Image:Circle area Monte Carlo integration2.svg. Other packages (lattice, ggplot2) provide alternative graphics facilities or syntax. Here is another example with data.
cluster heat map: where magnitudes are laid out into a matrix of fixed cell size whose rows and columns are categorical data. For example, the graph to the right. spatial heat map: where no matrix of fixed cell size for example a heat-map. For example, a heat map showing population densities displayed on a geographical map; Stripe graphic ...
Each control 'point' is actually a full vector of control points, defining a curve. These curves share their degree and the number of control points, and span one dimension of the parameter space. By interpolating these control vectors over the other dimension of the parameter space, a continuous set of curves is obtained, defining the surface.