Spectral characterizations of two families of nearly complete bipartite graphs

TL;DR

This paper investigates spectral properties of nearly complete bipartite graphs, proving some are uniquely determined by their spectra while others are not, based on modifications to complete bipartite graphs.

Contribution

It demonstrates that removing an edge from a complete bipartite graph yields a spectrum-determined graph, while certain vertex and edge additions do not.

Findings

Removing an edge from a complete bipartite graph results in a spectrum-determined graph.

Adding a vertex and an incident edge to a complete bipartite graph can produce graphs not determined by their spectra.

Some nearly complete bipartite graphs are uniquely identified by their spectra, others are not.

Abstract

It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each of which is obtained from a complete bipartite graph by adding a vertex and an edge incident on the new vertex and an original vertex, which are not determined by their spectra.