Path-tables of trees: a survey and some new results

Authors

Kevin Asciak, University of MaltaFollow

Publication Date

2015

Abstract

The (vertex) path-table of a tree Τ contains quantitative information about the paths in Τ. The entry (i,j) of this table gives the number of paths of length j passing through vertex ^vi. The path-table is a slight variation of the notion of path layer matrix. In this survey we review some work done on the vertex path-table of a tree and also introduce the edge path-table. We show that in general, any type of path-table of a tree Τ does not determine Τ uniquely. We shall show that in trees, the number of paths passing through edge xy can only be expressed in terms of paths passing through vertices x and y up to a length of 4. In contrast to the vertex path-table, we show that the row of the edge path-table corresponding to the central edge of a tree Τ of odd diameter, is unique in the table. Finally we show that special classes of trees such as caterpillars and restricted thin trees (RTT) are reconstructible from their path-tables.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Asciak, Kevin (2015) "Path-tables of trees: a survey and some new results," Theory and Applications of Graphs: Vol. 2: Iss. 2, Article 4.
DOI: 10.20429/tag.2015.020204
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol2/iss2/4