On the Inverse of Bipartite Graphs with Unique Perfect Matchings and Reciprocal Eigenvalue Properties

Abstract

The study of graph structures via different properties of its adjacency matrix is a widely studied subjects. Sometimes different structural properties of a graph get characterized by different properties of the eigenvalues and the eigenvectors of the associated adjacency matrix. For example, it is well-known that ‘a (simple) graph G is connected if and only if (G) (spectral radius) is a simple eigenvalue with an eigenvector whose coordinates are nonzero and of the same sign’. There are many interesting results exhibiting the relationship of the graph structure with the eigenvalues and eigenvectors. This thesis aims to establish such relationships with regard to the concepts inverse graph and reciprocal eigenvalue properties. Both of these play important roles in quantum chemistry.