Recent progress on graphs with fixed smallest eigenvalue

2020/11/24 by Koolen, Jack H., Cao, Meng-Yue, Yang, Qianqian
#05C75 #05D99 #05E30 #11H06 #Combinatorics (math.CO) #FOS: Mathematics #Primary 05C50. Secondary 05C22

paper · doi:10.48550/arxiv.2011.11935

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.

Related

• A structure theory for regular graphs with fixed smallest eigenvalue
• A generalization of a theorem of Hoffman
• The distance-regular graphs such that all of its second largest local eigenvalues are at most one
• The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters
• A Bose-Laskar-Hoffman theory for μ-bounded graphs with fixed smallest eigenvalue