Arrowhead and Diamond Diameters

TL;DR

This paper analyzes the diameters and antipodal distributions of Arrowhead and Diamond hierarchical Cayley graphs on the triangular grid, providing explicit formulas and characterizations for both oriented and non-oriented cases.

Contribution

It derives explicit diameter formulas and characterizes antipodal distributions for Arrowhead and Diamond graphs, clarifying their structural properties.

Findings

Diameters are explicitly expressed for both graph types.

Antipodal distributions are fully characterized.

Results apply to both oriented and non-oriented versions.

Abstract

{\em Arrowhead} and {\em diamond} are two hierarchical Cayley graphs defined on the triangular grid. In their undirected version, they are isomorphic and merely define two distinct representations of the same graph. This paper gives the expression of their diameter, in the oriented and non--oriented case. It also displays the full distribution of antipodals.