We study graph homomorphisms over finite graphs from a viewpoint of reproducing kernel Hilbert space theory. In particular, introducing de Branges-Rovnyak theory into graph theory, the relation between injective graph homomorphisms and de Branges-Rovnyak spaces is discussed in detail.
"Gram Matrices of Reporducing Kernel Hilbert Spaces over Graphs II (Graph Homomorphisms and De Branges-Rovnyak Spaces)." Nihonkai Math. J. 26 (1) 15 - 29, 2015.