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The line graph of a bipartite graph is perfect (see Kőnig's theorem), but need not be bipartite as the example of the claw graph shows. The line graphs of bipartite graphs form one of the key building blocks of perfect graphs, used in the proof of the strong perfect graph theorem. [15]
In linear algebra, a linear function is a map f between two vector spaces such that. Here a denotes a constant belonging to some field K of scalars (for example, the real numbers) and x and y are elements of a vector space, which might be K itself. In other terms the linear function preserves vector addition and scalar multiplication .
Line chart. A line chart or line graph, also known as curve chart, [ 1] is a type of chart that displays information as a series of data points called 'markers' connected by straight line segments. [ 2] It is a basic type of chart common in many fields. It is similar to a scatter plot except that the measurement points are ordered (typically by ...
The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding section. If b = 0, the line is a vertical line (that is a line parallel to ...
A linear function is a polynomial function in which the variable x has degree at most one: [2] . Such a function is called linear because its graph, the set of all points in the Cartesian plane, is a line. The coefficient a is called the slope of the function and of the line (see below). If the slope is , this is a constant function defining a ...
A graph with six vertices and seven edges. In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called vertices (also called nodes or points) and each of the related pairs of vertices ...
Linear interpolation on a set of data points (x0, y0), (x1, y1), ..., (xn, yn) is defined as piecewise linear, resulting from the concatenation of linear segment interpolants between each pair of data points. This results in a continuous curve, with a discontinuous derivative (in general), thus of differentiability class .
A prototypical example that gives linear maps their name is a function ::, of which the graph is a line through the origin. [ 7 ] More generally, any homothety v ↦ c v {\textstyle \mathbf {v} \mapsto c\mathbf {v} } centered in the origin of a vector space is a linear map (here c is a scalar).
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