Study of Joining and Spectral Radius of Hypergraphs Using Equitable Partition of Their Vertex Sets

Sarkar, Amitesh (2023) Study of Joining and Spectral Radius of Hypergraphs Using Equitable Partition of Their Vertex Sets. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Amitesh Sarkar (16RS059)) 16RS059.pdf - Submitted Version Restricted to Repository staff only Download (31MB)

Abstract

Here, we consider a matrix representation of hypergraphs and investigate their spectra. In order to determine the eigenvalues of various hypergraphs, we extend the notion of equitable partition and joining operation for hypergraphs. This joining operation has been applied in a number of contexts, including the construction of a hypergraph of hypergraphs and the (un)weighted joining of a set of (un)weighted hypergraphs. Using this joining operation, we also constructed the corona of hypergraphs, namely the vertex corona and edge corona. We find the spectrum of hypergraphs created through our joining operation. As an application of this operation and equitable partition, we derive the characteristic polynomial of a complete m-uniform m-partite hypergraph. We generalize a method, similar to the Godsil-McKay switching to construct a pair of non-isomorphic co-spectral hypergraphs. Furthermore, we explain how to create an infinite number of pairs of non-isomorphic co-spectral hypergraphs using vertex corona. By examining the edge corona, we find the full spectrum of m-uniform s-loose cycles for m > 2s and the characteristic polynomial of an m-uniform S-loose path. Additionally, we derive several eigenvalues of m-uniform s-loose path. We also focus on the maximal moduli of the eigenvalues, i.e., the spectral radii of a few linear hypergraphs related to the adjacency matrices. For this purpose, we have introduced another hypergraph operation, namely, the edge spreading, and see how it affects the spectral radius of the newly produced hypergraph. We identify the hypertrees with the maximum and second-maximum spectral radii among all those with a particular diameter. We determine the seven hypertrees with the first seventh biggest spectral radii. Unicyclic hypergraphs having the largest, second-largest, and third-largest spectral radii with a constant cycle length are also identified. The highest and second-highest spectral radii of the bicyclic linear hypergraphs are also discovered. We have also considered two types of tricyclic linear hypergraphs and identified the tricyclic hypergraphs having the first two largest spectral radii of both types.

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