In this paper, we introduce the notion of reproducing kernel Hilbert spaces for graphs and the Gram matrices associated with them. Our aim is to investigate the Gram matrices of reproducing kernel Hilbert spaces. We provide several bounds on the entr
ies of the Gram matrices of reproducing kernel Hilbert spaces and characterize the graphs which attain our bounds.
In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least -1-tau, where tau is the golden ratio, can be described by a finite set of fat (-1-tau)-irreducible Hoffman graphs. In the terminology of Woo and Neumaier, we mean t
hat every fat Hoffman graph with smallest eigenvalue at least -1-tau is an H-line graph, where H is the set of isomorphism classes of maximal fat (-1-tau)-irreducible Hoffman graphs. It turns out that there are 37 fat (-1-tau)-irreducible Hoffman graphs, up to isomorphism.
