Neighborhood Champions in Regular Graphs

For a vertex \( x \) in a graph \( G \), we define \( \Psi_1(x) \) to be the number of edges in the closed neighborhood of \( x \). Vertex \( x^* \) is a neighborhood champion if \( \Psi_1(x^*) > \Psi_1(x) \) for all \( x \neq x^* \). We also refer to such an \( x^* \) as a unique champion. For \( d \geq 4 \), let \( n_0(1,d) \) be the smallest number such that for every \( n \geq n_0(1,d) \) there exists an \( n \)-vertex \( d \)-regular graph with a unique champion. Our main result is that \( n_0(1,d) \) satisfies \( d+3 \leq n_0(1,d) < 3d+1 \). We also observe that there can be no unique champion vertex when \( d = 3 \).