On graphs with just three distinct eigenvalues

Abstract

Let G be a connected non-bipartite graph with exactly three distinct eigenvalues ρ, µ, λ, where ρ > µ > λ. In the case that G has just one non-main eigenvalue, we find necessary and sufficient spectral conditions on a vertex-deleted subgraph of G for G to be the cone over a strongly regular graph. Secondly, we determine the structure of G when just µ is non-main and the minimum degree of G is 1 + µ − λµ: such a graph is a cone over a strongly regular graph, or a graph derived from a symmetric 2-design, or a graph of one further type.

More on graphs with just three distinct eigenvalues

“…It is an open problem to determine all the graphs in C 1 , and another open problem to determine all the graphs in C 2 . 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].…”

“…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].…”

“…In [14] it was noted that any graph G in C 1 ∩ C 2 is either integral or complete bipartite. Examples include the following, where G has spectrum ρ, µ (k) , λ (l) with ρ > µ > λ.…”

“…(1) G is of conical type: here G is the cone over a strongly regular graph with parameters λ 2 µ + λ 2 − λµ, µ − λµ, 2µ + λ, µ (see [12,14]);…”

Graphs with two main and two plain eigenvalues

“…Rowlinson [14] characterized the strongly biregular graphs among the graphs with three distinct eigenvalues. For the convenience of the reader, we include a proof of it.…”