On Extension of Regular Graphs and Adjacency Matrix of Hypergraphs

Bej, Saptarshi (2014) On Extension of Regular Graphs and Adjacency Matrix of Hypergraphs. Masters thesis, Indian Institute of Science Education and Research Kolkata.

PDF (MS dissertation of Saptarshi Bej (09MS012)) Saptarshi_Bej_(09MS012).pdf - Submitted Version Restricted to Repository staff only Download (493kB)

Abstract

In the present work, we studied several cases for knowing when a regular graph can be extended to another regular graph of higher regularity by only adding edges. Though this problem is equivalent to finding matchings in the complement of a regular graph, yet depending on number of vertices and regularity we can come up with some interesting results. Hypergraphs have also been studied far and widely. But defining Adjacency matrices is always tricky. One way has been defined by Nufellen in his paper. Here we try to find a new definition of Adjacency matrix of hypergraphs so that it exhibits some interesting properties as it does in case of general graphs. In the second chapter we will talk mainly of the introductory definitions that are required to fully understand the problems that has been discussed. In the third chapter we will discuss several cases where we can or can’t extend an r-regular graph to r+1 regular graph. We will also see several results on existence of extendable regular graphs. In the fourth chapter we will discuss about the Adjacency Matrix of Hypergraphs and some of its basic properties.

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