In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Coates graph or Coates flow graph, named after C.L. Coates, is a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
associated with the Coates' method for the solution of a
system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three ...
.
[
The Coates graph ''G''''c''(A) associated with an ''n'' × ''n'' ]matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
A is an ''n''-node, weighted, labeled, directed graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
Definition
In formal terms, a directed graph is an ordered pa ...
. The nodes, labeled 1 through ''n'', are each associated with the corresponding row/column of A. If entry ''a''''ji'' ≠ 0 then there is a directed edge from node ''i'' to node ''j'' with weight ''a''''ji''.[ In other words, the Coates graph for matrix A is the one whose ]adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simp ...
is the transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of A.
See also
* Flow graph (mathematics)
A flow graph is a form of digraph associated with a set of linear algebraic or differential equations:
:"A signal flow graph is a network of nodes (or points) interconnected by directed branches, representing a set of linear algebraic equations. T ...
* Mason graph
A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized flow graph, a directed graph in which nodes represent system variables, ...
References
{{reflist, refs=
[{{cite book , author=Wai-Kai Chen , year=1976 , title=Applied Graph Theory , chapter=The associated Coates graph , publisher=North Holland Publishing Company , page=142 , url=https://www.google.com/search?tbm=bks&q=%22For+a+square+matrix%22+%22is+an+n-node,+weighted,+labeled,+directed+graph%22]
[{{cite journal , last=Coates , first=C.L. , year=1959 , title=Flow-graph solutions of linear algebraic equations , journal=IRE Trans. Circuit Theory , doi=10.1109/TCT.1959.1086537 , volume=CT-6 , issue=2 , pages=170–187]
[{{cite book , last1=Thulasiraman , first1=K. , last2=Swamy , first2=M. N. S. , year=1992 , title=Graphs:Theory and Algorithms , chapter=§6.11 The Coates and Mason graphs , isbn=0-471-51356-3 , pages=163–169 , publisher=John Wiley & Sons , chapter-url=https://books.google.com/books?id=4kHN6uSinQoC&q=%22coates+method%22&pg=PA163]
Application-specific graphs
Linear algebra