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Is there a general method that can be used find the diameter of a n-cube? In particular what if I want to find the diameter of a 4-cube can someone suggest me a method or hint. I would much appreciate it. Thanks

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2 Answers

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The diameter of a set is the distance between the two points that are farthest apart (more or less, with some "supremum" stuff for more complicated objects). For a standard unit cube in $n$-dimensions, the vertices are points $(x_1, \ldots, x_n)$, where each $x_i$ is either $0$ or $1$, and two points at the opposite ends of a diagonal in this situation are $(0, 0, \ldots, 0)$ and $(1, 1, \ldots, 1)$. The distance between these is $$\sqrt{ (1-0)^2 + \ldots (1-0)^2} = \sqrt{n}.$$

So the diameter of an $n$-cube whose side has length $1$ is $\sqrt{n}$.

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The diameter of an $n$-cube graph is $n$. For let $v$ be a vertex of $Q_n$. Since $Q_n$ is vertex-transitive, it doesn't matter which $v$ we choose, so say it is the one whose $n$ coordinates are all zeroes. Then $v'$, the farthest vertex from $v$, is clearly the one whose $n$ coordinates are all ones, at a distance of $n$.

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