Recent progress on graphs with fixed smallest eigenvalue

TL;DR

This survey reviews recent advances in the study of graphs with fixed smallest eigenvalue, focusing on Hoffman graphs, distance-regular graphs, co-edge regular graphs, and signed graphs, highlighting theoretical developments and characterizations.

Contribution

It provides a comprehensive overview of recent progress and new findings in the spectral graph theory related to fixed smallest eigenvalues, including applications and characterizations.

Findings

Hoffman graphs theory and applications

Characterizations of distance-regular graphs with fixed eigenvalues

New results on signed graphs with fixed smallest eigenvalue

Abstract

We give a survey on graphs with fixed smallest eigenvalue, especially on graphs with large minimal valency and also on graphs with good structures. Our survey mainly consists of the following two parts: (i) Hoffman graphs, the basic theory related to Hoffman graphs and the applications of Hoffman graphs to graphs with fixed smallest eigenvalue and large minimal valency; (ii) recent results on distance-regular graphs and co-edge regular graphs with fixed smallest eigenvalue and the characterizations of certain families of distance-regular graphs. At the end of the survey, we also discuss signed graphs with fixed smallest eigenvalue and present some new findings.