Antipodal Graph

Define an antipode of a given graph vertex  as a vertex  at greatest possible graph distance from . Then an antipodal graph is a connected graph in which each vertex has exactly one antipode (Gorovoy and Zmiaikou 2021).

The numbers of antipodal graphs on , 2, ... nodes are 1, 1, 0, 2, 1, 5, 8, 73, 660, 9909, ... (OEIS A349635).

Examples of antipodal graphs include Bruhat graphs, cocktail party graphs , even cycle graphs  (Gorovoy and Zmiaikou 2021), hypercube graphs , and path graphs of odd length  (Gorovoy and Zmiaikou 2021).

A tree is antipodal iff it has a unique longest path and this path is of odd length (Gorovoy and Zmiaikou 2021).

Every vertex of a geodetic Hamiltonian graph has at least two antipodes (Gorovoy and Zmiaikou 2021).

REFERENCES

Gorovoy, D. and Zmiaikou, D. "On Graphs with Unique Geoodesics and Antipodes." 19 Nov 2021. https://arxiv.org/abs/2111.09987.

Sloane, N. J. A. Sequence A349635 in "The On-Line Encyclopedia of Integer Sequences."