Walk-Based Transfinite Graphs and Networks

  • Armen H. Zemanian


The theory of transfinite graphs developed so far has been based on the ideas that connectedness is accomplished through paths and that the infinite extremities of the graph are specified through equivalence classes of one-ended paths. This is a natural extension of finite graphs because connectedness for finite graphs is fully characterized by paths; indeed, any walk terminating at two nodes of a finite graph contains a path doing the same. However, such is no longer the case for transfinite graphs. Indeed, path-connectedness need not be transitive as a binary relationship among transfinite nodes, and Condition 3.1-2 was imposed to ensure such transitivity. Without that condition, distances as defined by paths do not exist between certain pairs of nonsingleton nodes. This limitation is also reflected in the theory of transfinite electrical networks by the fact that node voltages need not be uniquely determined when they are defined along paths to a chosen ground node.


High Rank Current Vector Equivalence Relationship Finite Graph Node Voltage 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Armen H. Zemanian
    • 1
  1. 1.Department of Electrical EngineeringState University of New York at Stony BrookStony BrookUSA

Personalised recommendations