Biregular graphs with three eigenvalues

Abstract

Dedicated to Misha Klin on the occasion of his retirement.Abstract. We consider nonregular graphs having precisely three distinct eigenvalues. The focus is mainly on the case of graphs having two distinct valencies and our results include constructions of new examples, structure theorems, valency constraints, and a classification of certain special families of such graphs. We also present a new example of a graph with three valencies and three eigenvalues of which there are currently only finitely many known examples.

On graphs with just three distinct eigenvalues

“…(Equations (7) and (8) are special cases of [3,Theorem 4.3(iv)]. ) Now we can find k and l from the equations ρ + kµ + lλ = 0, 1 + k + l = n. We obtain…”

“…A wider discussion of the biregular graphs in C 1 may be found in the recent paper [3]. Here we shall also make use of the following intermediate result.…”

Graphs with two main and two plain eigenvalues

“…A connected regular graph with at most three distinct eigenvalues is known to be strongly regular, and see, for example [3] for a survey on strongly regular graphs. The connected non-regular graphs with three distinct eigenvalues have been studied by, for example, De Caen, Van Dam & Spence [7], Bridges & Mena [1], Muzychuk & Klin [13], Van Dam [17] and Cheng, Gavrilyuk, Greaves & Koolen [6]. The connected regular graphs with four distinct eigenvalues were studied by Van Dam [18], Van Dam & Spence [19] and Huang & Huang [12], among others.…”

More on graphs with just three distinct eigenvalues

“…From the equations 15 + 3k − 3l = 0, 1 + k + l = 39, we obtain the contradiction 6 + 2k = 39. This non-existence result can also be extracted from the results in [3].…”

“…Here we continue the investigation of graphs in C 1 ∩ C 2 begun in [14]. Independently the authors of [3] investigated the biregular graphs in C 1 , and it is not difficult to see that these are precisely the graphs in C 1 ∩ C 2 : this follows from [3, Theorem 4.3(i)] and [14,Lemma 2.2]. Some examples of biregular graphs in C 2 \ C 1 (of order 16) can be found in [10, Table 1].…”