NDLI: Walk-Based Transfinite Graphs and Networks

Content Provider	Springer Nature Link
Author	Zemanian, A.H.
Copyright Year	2004
Abstract	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 one-way infinite paths. As a result, a variety of difficulties arise in that theory, leading to the need to restrict such path-based graphs in various ways to obtain certain results. In this work, we present a more general theory of transfinite graphs, wherein connectedness and the designation of extremities are accomplished through walks rather than paths. This leads to a simpler and yet more general theory, wherein new kinds of transfinite extremities are also encompassed. For instance, an ordinal-valued distance function can now be defined on all pairs of walk-connected nodes, in contrast to the path-based theory, wherein no distance function is definable for those pairs of nodes that are not path connected even though they are walk connected. Some results concerning eccentricities, radii, and diameters are presented in this more general walk-based graph theory. Another new result herein is the development of an electrical network theory for networks whose graphs are walk based. A unique voltage-current regime is established under certain conditions. The current regime is built up from current flows in closed transfinite walks—in contrast to a prior theory based upon flows in transfinite loops. A notable advantage of the present approach is that node voltages with respect to a given ground node are always unique whenever they exist. The present approach is more general in that it provides nontrivial voltage-current regimes for certain networks for which the prior approach would only provide trivial solutions having only zero currents and voltages everywhere.
Starting Page	1
Ending Page	31
Page Count	31
File Format	PDF
ISSN	0278081X
Journal	Circuits, Systems, and Signal Processing
Volume Number	23
Issue Number	1
e-ISSN	15315878
Language	English
Publisher	Birkhäuser-Verlag
Publisher Date	2004-01-01
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Signal Processing

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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