
8.2 Concepts of Path and Path Expressions

A graph is a representation of nodes and links. Every link of a graph can be labelled with lower case letter or sequence of letters which is referred to as link weights. Link weights denote the sequence of dataflow actions on that link.

The path segment or tracing a path involves traversing a successive links names from start node to end node in that path segment. Path segment is represented by capital case letter. If a same path segment is traced more than one time, the number of times such path segments are traced is represented as a power of that path segment.

The path product is a concatenation of two or more successive path segments.
The path sum represents a set of parallel paths that exist between two nodes in a graph.
The Path expression is a representation of a set of all possible paths between two nodes.
Following sections provide examples on the above described concepts.

8.2.1 Examples on Path Products

Path product is the concatenation of two or more successive path segments.
Consider the following graph that has nodes 1, 2, 3 with links labeled as a and b as depicted below:


Here link a and link b are two successive links. The path between node 1 and node 3 is represented by path product as ‘ab’.
Let us consider the following graph with path segments X, Y and Z defined as ;
X = abc
Y = defg
Z= klmn


In the above graph, following are few examples of path products:
Then the path between node 1 to node 3 is represented by XY, that is, path product corresponding to X followed by Y. This is denoted by
XY = abcdefg
Similarly,
hY = hdefg
Yi = defgi
XYZ= abcdefgklmn
hYi= habci
XYi = abcdefgi
XYZj= abcdefgklmnj

8.2.2 Examples on Path Sum

The path sum indicates that when there are many parallel paths between the two nodes of interest, one can choose any one of them to start from the initial node and to reach the final node in that path segment. The path sum is represented by the “+” sign which is a “or” operation.

Consider the following graph that has nodes 1, 2 with parallel links labeled as a and b as depicted below:



Here link a and link b are two parallel links. The path between node 1 and node 2 is represented by path sum as ‘a + b’.
Consider another graph with path segments X, Y and Z defined as ;
X = abc
Y = defg
Z= klmn




In this graph, the path sum between node and node 2 is represented Notationally as:
X+h= abc + h






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