gives the edge cycle matrix of a graph g.
uses rules vw to specify the graph g.
- EdgeCycleMatrix is also known as tie-set matrix or loop matrix.
- EdgeCycleMatrix returns a matrix cij where each row corresponds to a cycle i in the graph g, and each column corresponds to an edge ej.
- For an undirected graph, cij is 1 if edge ej is part of cycle i and zero otherwise.
- For a directed graph, cij is 1 if edge ej is part of cycle i, -1 if edge ej in reverse direction is part of cycle i, and zero otherwise.
- Edge ej is the edge at position j in EdgeList[g], and the index j for an edge ej can be found from EdgeIndex[g,ej].
- EdgeCycleMatrix gives a basis for all the cycles in the graph g.
- EdgeCycleMatrix works with undirected graphs, directed graphs, multigraphs, and mixed graphs.
Background & Context
- EdgeCycleMatrix returns a matrix cij in which each row corresponds to a cycle i in a graph and each column corresponds to an edge ej. An edge cycle matrix is determined by the incidences of edges and cycles in a graph, and cycles in an edge cycle matrix form a cycle basis of a graph. Cycle bases are useful in the study of chemical graphs, to generate large cycle families, and to compute voltage or current in a circuit. Edge cycle matrices are also known as tie-set or loop matrices.
- For an undirected graph, cij is 1 if edge ej is part of cycle i and zero otherwise. For a directed graph, cij is 1 if edge ej is part of cycle i, if edge ej in reverse direction is part of cycle i, and zero otherwise.
- FindFundamentalCycles is a related function that can be used to return a list of fundamental cycles of a graph.
Examplesopen allclose all
Basic Examples (2)
EdgeCycleMatrix works with undirected graphs:
Use rules to specify the graph:
EdgeCycleMatrix works with large graphs:
Properties & Relations (3)
Use EdgeList to obtain a cycle representation:
A connected graph with n vertices and m edges:
Has an edge cycle matrix of dimensions (m-n+1)×m:
Compute the EdgeCycleMatrix from the IncidenceMatrix:
Compare with EdgeCycleMatrix:
Wolfram Research (2014), EdgeCycleMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/EdgeCycleMatrix.html (updated 2015).
Wolfram Language. 2014. "EdgeCycleMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/EdgeCycleMatrix.html.
Wolfram Language. (2014). EdgeCycleMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EdgeCycleMatrix.html