On split graphs with four distinct eigenvalues

TL;DR

This paper classifies connected bidegreed 3-extremal split graphs, which are graphs with diameter 3 and exactly four eigenvalues, and explores constructions of non-bidegreed variants.

Contribution

It provides a complete classification of connected bidegreed 3-extremal split graphs and methods to construct non-bidegreed examples.

Findings

Complete classification of connected bidegreed 3-extremal split graphs

Construction methods for non-bidegreed 3-extremal split graphs

Insights into the spectral properties of split graphs with four eigenvalues

Abstract

It is a well-known fact that a graph of diameter $d$ has at least $d+1$ eigenvalues. Let us call a graph \emph{$d$-extremal} if it has diameter $d$ and exactly $d+1$ eigenvalues. Such graphs have been intensively studied by various authors. %Much attention has been devoted to the study of graphs that are extremal with respect to this relation: \emph{i.e} have diameter $d$ and exactly $d+1$ distinct eigenvalues. A graph is \emph{split} if its vertex set can be partitioned into a clique and a stable set. Such a graph has diameter at most $3$. We obtain a complete classification of the connected bidegreed $3$-extremal split graphs. We also show how to construct certain families of non-bidegreed $3$-extremal split graphs.