Shilla distance-regular graphs

A Shilla distance-regular graph (say with valency k) is a distance-regular graph with diameter 3 such that its second largest eigenvalue equals to a3. We will show that a3 divides k for a Shilla distance-regular graph , and for we define b = b() := k a3 . In this paper we will show that there are finitely many Shilla distance-regular graphs with fixed b() 2. Also, we will classify Shilla distance-regular graphs with b() = 2 and b() = 3. Furthermore, we will give a new existence condition for distance-regular graphs, in general. Key Words: distance-regular graph; Existence condition; Terwilliger graph 2000 Mathematics Subject Classification: 05E30

• Type: Journal
• Year: 2010
• Where: EJC
• Authors: Jack H. Koolen, Jongyook Park
• Keywords: Distance-regular Graph, EJC 2007, EJC 2010, Largest Eigenvalue, Shilla Distance-regular Graphs