Cores and Independence Numbers of Grassmann Graphs

Abstract

A graph G is a core if every endomorphism of G is an automorphism. Let $$J_q(n,m)$$ be the Grassmann graph with parameters q, m, n. We prove that many Grassmann graphs are cores, and both $$J_2(2k,2)$$ and $$J_q(2^k,2)$$ are not cores. We also obtain the independence number of $$J_q(n,2)$$ . In further to study cores and coding theory, it is important to estimate the upper bound of the independence number of $$J_q(n,m)$$ . Using a vertex-transitive subgraph of $$J_q(n,m)$$ , we obtain upper bounds on the independence number of $$J_q(n,m)$$ , which are also an improvement of bounds for the size of constant dimension codes in a 2011 paper of Etzion and Vardy.