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The Number of Paths and Cycles in a Digraph

Published online by Cambridge University Press:  01 January 2025

Dorwin Cartwright  and
Terry C. Gleason
Dorwin Cartwright
Affiliation:
University of Michigan
Terry C. Gleason
Affiliation:
University of Michigan
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Abstract

An algorithm is presented for constructing from the adjacency matrix of a digraph the matrix of its simple n-sequences. In this matrix, the i, j entry, i ≠j, gives the number of paths of length n from a point vi to a point vj; the diagonal entry i, i gives the number of cycles of length n containing vi. The method is then generalized to networks—that is, digraphs in which some value is assigned to each line. With this generalized algorithm it is possible, for a variety of value systems, to calculate the values of the paths and cycles of length n in a network and to construct its value matrix of simple n-sequences. The procedures for obtaining the two algorithms make use of properties of a line digraph—that is, a derived digraph whose points and lines represent the lines and adjacency of lines of the given digraph.

Type
Original Paper
Information
Psychometrika , Volume 31 , Issue 2 , June 1966 , pp. 179 - 199
Copyright
Copyright © 1966 Psychometric Society

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Footnotes

*

The research reported here was supported by Grant NSF-G-17771 from the National Science Foundation. We wish to thank Professor Frank Harary for suggesting certain ways of improving an earlier draft of this paper.

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