In
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.
[
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'' (franchi ...
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 pai ...
. 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 simple ...
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
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