## Doctoral Theses

### External Graph Theoretic Problems with Applications to Communication Networks.

2-28-1966

2-28-1967

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Computer Science

#### Department

Research and Training School (RTS)

#### Abstract (Summary of the Work)

Graphs havre now become recognized models for a wide variety of situations. Whenever we have a collection of è´¸bjects with a binary relation defined on them graphs serve as excellopt toola to study the combinatoria properties of the collection with respect to the binary relation. D. Konig was the firat person to recognize the usefulness of graph theoretic modols. He conceived of a unified study of grapha under an abstract set up. His book was a pioneering work in this field.The graph theoretic problems embodied in this thesis have been motivated, by situations arising in communicationhave been motivated, by situations arising in communication networks. In terms of graphs these problems ask for the extremal structures wi th preassigmed diameters and their variations under suppression of vertices and edges. The motivation for the problems and their applications is deferred to the penultimate chapter. This has been found reasonable, in any case not disorderly, as the appropriate combinatorial problems in terms of graphs seem to be of great interest on their own. Perhang we have the o to study the extremal structures of graphs satisfy given property and retaining the same property after portions of the graphs have been suppressed.All the graph theoretic problems considered her round the distance metric defined for graphs. An at made in the last chapter to define a distance between two columns of a (0, 1)-matrix. In light of this it to be possible to carry the analogy from graphs to mThe contributions of this thesis have been divided into six chapters and an appendix. At the beginning each chapter a detailed summary of the results in the cháº¡pter is provided. Berge [ 1] has been followed proved that if the number of vertices of graph is suffici- ently large compared to the stipulated number of suppressible vertices and if the diameter should remain two throughout, then the extremal graph will be an appropriate complete bipartite graph. This is followed by the results concerning the extremal structures when the diameter is allowed to vary from 2 to ((2 3) wunder the suppression of a single Several upper bounds have been given and some vertex. conjectures have been made about the extremal numbers that have not been determined here.In chapter 2 problems similar to those in chapter l are considered with regard to edge suppression. Here again the most general result is when the diameter remains two throughout, We prove that the extrenal structures in this case are obtained by completing one of the aets of thecase are obtained by completing one of the sets of the extremal complete bipartite graph of chapter 1. As in chapter 1, atruotures for other diame tral variations hnve also been studied and some conjectures presented. The dual opy of C 4- aspects of problems of chapter 1 and problems of chapte are disoussed in the appendix.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842929

ISILib-TH8