Graph Cores via Universal Completability

Abstract A framework for a graph G = ( V , E ) , denoted G ( p ) , consists of an assignment of real vectors p = ( p 1 , p 2 , … , p | V | ) to its vertices. A framework G ( p ) is called universally completable if for any other framework G ( q ) that satisfies p i T p j = q i T q j for all i = j and ( i , j ) ∈ E there exists an isometry U such that U q i = p i for all i ∈ V . A graph is called a core if all its endomorphisms are automorphisms. In this work we identify a new sufficient condition for showing that a graph is a core in terms of the universal completability of an appropriate framework for the graph. To use this condition we develop a method for constructing universally completable frameworks based on the eigenvectors for the smallest eigenspace of the graph. This allows us to recover the known result that the Kneser graph K n : r and the q -Kneser graph q K n : r are cores for n ≥ 2 r + 1 . Our proof is simple and does not rely on the use of an Erdős-Ko-Rado type result as do existing proofs. Furthermore, we also show that a new family of graphs from the binary Hamming scheme are cores, that was not known before.