Recent Progress on Graphs with Fixed Smallest Eigenvalue
Recent progress on graphs with fixed smallest eigenvalue Jack H. Koolena,b, Meng-Yue Caoc, and Qianqian Yang∗a aSchool of Mathematical Sciences, University of Science and Technology of China, 96 Jinzhai Road, Hefei, 230026, Anhui, PR China. bCAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui, 230026, PR China cSchool of Mathematical Sciences, Beijing Normal University, 19 Xinjiekouwai Street, Beijing, 100875, PR China. November 25, 2020 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 arXiv:2011.11935v1 [math.CO] 24 Nov 2020 present some new findings. 1 Introduction All graphs mentioned in this paper are finite, undirected and simple. For undefined notations see [13], [35] and [43]. Unless we specify a different matrix, by an eigenvalue of a graph, we mean ∗Corresponding author. 2010 Mathematics Subject Classification. Primary 05C50. Secondary 05C22, 05C75, 05E30, 05D99, 11H06. E-mail addresses: koolen@ustc.edu.cn (J.H. Koolen), cmy1325@163.com (M.-Y. Cao), qqyang91@ustc.edu.cn (Q.
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