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. The nodes in a flow graph are used to represent the variables, or parameters, and the connecting branches represent the coefficients relating these variables to one another. The flow graph is associated with a number of simple rules which enable every possible solution [related to the equations] to be obtained."
For flow graph in computer science, see Control-flow graph, Rooted graph § Flow graphs, and Flowchart.

Although this definition uses the terms "signal-flow graph" and "flow graph" interchangeably, the term "signal-flow graph" is most often used to designate the Mason signal-flow graph, Mason being the originator of this terminology in his work on electrical networks. Likewise, some authors use the term "flow graph" to refer strictly to the Coates flow graph. According to Henley & Williams:

"The nomenclature is far from standardized, and...no standardization can be expected in the foreseeable future."

A designation "flow graph" that includes both the Mason graph and the Coates graph, and a variety of other forms of such graphs appears useful, and agrees with Abrahams and Coverley's and with Henley and Williams' approach.

A directed network – also known as a flow network – is a particular type of flow graph. A network is a graph with real numbers associated with each of its edges, and if the graph is a digraph, the result is a directed network. A flow graph is more general than a directed network, in that the edges may be associated with gains, branch gains or transmittances, or even functions of the Laplace operator s, in which case they are called transfer functions.

There is a close relationship between graphs and matrices and between digraphs and matrices. "The algebraic theory of matrices can be brought to bear on graph theory to obtain results elegantly", and conversely, graph-theoretic approaches based upon flow graphs are used for the solution of linear algebraic equations.

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