On antipodal properties for eigenfunctions of graphs

A graph $G=(V,E)$ of diameter $d$ is termed to be antipodal if for any vertex $x\in{V}$ there is precisely one another $x^\prime\in{V}$ such that $d(x,x^\prime)=d$. In addition, an antipodal graph is called rigid if for any pair of its antipodal vertices $x,x^\prime\in{V}$ and any third vertex $y\in{V}$ the equality $d(x,x^\prime)=d(x,y)+d(y,x^\prime)$ holds. In this paper eigenfunctions of rigid antipodal graphs are investigated. It is shown that every homogeneous eigenfunction of such a graph with odd diameter is determined uniquely from its values on vertices in two middle layers of the graph.