On fat Hoffman graphs with smallest eigenvalue at least -3

Abstract

We investigate fat Hoffman graphs with smallest eigenvalue at least −3, using their special graphs. We show that the special graph S(H) of an indecomposable fat Hoffman graph H is represented by the standard lattice or an irreducible root lattice. Moreover, we show that if the special graph admits an integral representation, that is, the lattice spanned by it is not an exceptional root lattice, then the special graph S − (H) is isomorphic to one of the Dynkin graphs A n , D n , or extended Dynkin graphsà n orD n .

On graphs with smallest eigenvalue at least −3 and their lattices

“…, f r } and n 1 = n 2 = · · · = n r = n. With the above notations, we can now state the result of Hoffman and Ostrowski. For a proof of it, see [9,Theorem 2.14]. We will give a closely related result in the next section.…”

“…Now we give definitions and preliminaries of Hoffman graphs. For more details, see [9], [12] and [19]. (i) Every vertex with label f is adjacent to at least one vertex with label s;…”

Graphs with least eigenvalue −2: Ten years on

“…Subsequently, Woo and Neumaier [160] identified β ≈ −2.4812, the smallest root of the polynomial x 3 + 2x 2 − 2x − 2, as the second largest limit point less than −2. Two further limit points studied in the literature are −1 − τ (see [78]) and −3 (see [61]). Yu [129] provided infinitely many examples of connected regular graphs with least eigenvalue in the interval (β, −2).…”

Problems on Graphs with Fixed Smallest Eigenvalue

“…Note that λ min (h (t) ) = −t and λ min (h (t,1) ) = −t−1− √ t 2 −2t+5 2 . Figure 1: the Hoffman graph h (4) and h (3,1) We need one more Hoffman graph. Let c n be the Hoffman graph obtained by attaching one fat vertex to n vertices of K n+1 .…”