The graphs with all but two eigenvalues equal to  $$\pm 1$$ ± 1

Abstract

We determine all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from ±1 and decide which of these graphs are determined by their spectrum. This includes the so-called friendship graphs, which consist of a number of edge-disjoint triangles meeting in one vertex. It turns out that the friendship graph is determined by its spectrum, except when the number of triangles equals sixteen.Computing det(Q + I) and det(Q − I) shows that Q has no eigenvalue −1, and Q has an eigenvalue 1 if and only if b = 2. In case b = 2 we can rewrite A aswith k ≥ 2. Thus we obtained the graphs of Case (ii). If ΓX has no edges and at least two vertices, then these two vertices have the same neighbors, contradiction. So |ΓX| = 1 and we find

The non-bipartite graphs with all but two eigenvalues in [–1, 1]

“…To be more clear, let G denote the family of connected non-bipartite graphs all of whose eigenvalues are contained [−1, 1] except the largest and smallest ones. As mentioned before, in this article, we will completely describe the family G. This generalizes the main result of [5].…”

“…In [4], van Dam and Spence described the connected bipartite graphs with all but two eigenvalues in {−1, 1}. Very recently, Cioabȃ et al [5] explicitly determined the connected non-bipartite graphs with all but two eigenvalues in {−1, 1}.…”

Graphs with many valencies and few eigenvalues

“…In this context, it is good to mention that Cioabȃ, Haemers, Vermette, and Wong [6] recently showed that all Friendship graphs except the one on 33 vertices are determined by the spectrum. These graphs also have four distinct eigenvalues of which two are simple, just like the non-regular graphs in a regular two-graph.…”

Spectral characterization of new classes of multicone graphs

“…It is claimed in [8] that conjecture is valid. In [7], it is proved that if Γ is any graph cospectral with F n (n = 16), then Γ ∼ = F n . Abdollahi and Janbaz [3] precented a proof in special case of this topic.…”