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Date: 3-8-2016
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Date: 6-5-2022
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A walk is a sequence ,
,
, ...,
of graph vertices
and graph edges
such that for
, the edge
has endpoints
and
(West 2000, p. 20). The length of a walk is its number of edges.
A -walk is a walk with first vertex
and last vertex
, where
and
are known as the endpoints. Every
-walk contains a
-graph path (West 2000, p. 21).
A walk is said to be closed if its endpoints are the same. The number of (undirected) closed -walks in a graph with adjacency matrix
is given by
, where
denotes the matrix trace. In order to compute the number
of
-cycles, all closed
-walks that are not cycles must be subtracted. Similarly, to compute the number
of graph paths, all
-walks that are not graph paths (because they contain redundant vertices) must be subtracted (cf. Festinger 1949, Ross and Harary 1952).
For a simple graph (which has no multiple edges), a walk may be specified completely by an ordered list of vertices (West 2000, p. 20).
A trail is a walk with no repeated edges.
Festinger, L. "The Analysis of Sociograms Using Matrix Algebra." Human Relations 2, 153-158, 1949.
Ross, I. C. and Harary, F. "On the Determination of Redundancies in Sociometric Chains." Psychometrika 17, 195-208, 1952.
West, D. B. Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 20-21, 2000.
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