On Peripheral Vertices in Graphs

The periphery of a graph G is the subgraph of G induced by the vertices in G whose eccentricity equals the diameter of G. Graphs that are peripheries of graphs with specified diameter are characterized. For a graph G, the antipodal graph A(G) of G is the graph having the same vertex set as G and edge set E(A(G)) = {uv ∣ u, v ∈ V(G) and d_G(u, v) = diam G}. For a graph G and positive integer n, the nth iterated antipodal graph An(G) of G is defined as the graph A(An−1(G)) where Al(G) is A(G) and A0(G) denotes G. It is shown that there exist nonnegative integers n₁ and n₂ such that n₁ < n₂ and $$ {{A}^{{{{n}_{1}}}}}(G) \cong {{A}^{{{{n}_{2}}}}}(G) $$. The smallest postitive integer k for which there exists an integer ℓ such that Aℓ(G) ≅ Aℓ+k(G) is called the antipodal period of G. It is shown that for every positive integer k there exists a graph with antipodal period k.