A Nonregular Analogue of Conference Graphs

Abstract

We prove some results on graphs with three eigenvalues, not all integral; these are a natural generalization of the strongly regular conference graphs. We derive a Bruck Ryser type condition and construct some (nonregular) examples. 1999Academic Press

Combinatorial Designs with Two Singular Values II. Partial Geometric Designs

“…Let us first consider now the case that p 3 is not incident to the blocks in B 10 . From the fact that M has rank two it follows that p 3 is incident to the blocks in B 01 and B 00 , and that o = ±l.…”

Graphs with four distinct Laplacian eigenvalues

“…It is well known that the class of all regular graphs with three distinct adjacency (Laplacian) eigenvalues coincides with the class of strongly regular graphs. For results on nonregular graphs with three adjacency eigenvalues, we refer the reader to [1,3,5,16]. Regular graphs with four adjacency (Laplacian) eigenvalues were studied in [6,10], and nonregular bipartite graphs with four adjacency eigenvalues were investigated in [8,9], through the study of the incidence graphs of some combinatorial designs.…”

A note on graphs whose signless Laplacian has three distinct eigenvalues

“…Regular graphs with three distinct A-eigenvalues (L-eigenvalues, Q-eigenvalues) are precisely strongly regular graphs and therefore graphs with three distinct eigenvalues can be considered as a generalization of strongly regular graphs. For results on graphs with few distinct A-eigenvalues, we refer the reader to [1,2,3,6,7,8,10,12] and on graphs with few distinct L-eigenvalues to [9,13]. In this paper, we investigate graphs with three distinct Q-eigenvalues and show that the largest Q-eigenvalue of a connected graph G is noninteger if and only if G = K n − e for n ≥ 4.…”