Abstract
We calculate the distribution with respect to the vacuum state of the distance-[Formula: see text] graph of a [Formula: see text]-regular tree. From this result we show that the distance-[Formula: see text] graph of a [Formula: see text]-regular graphs converges to the distribution of the distance-[Formula: see text] graph of a regular tree. Finally, we prove that, properly normalized, the asymptotic distributions of distance-[Formula: see text] graphs of the [Formula: see text]-fold free product graph, as [Formula: see text] tends to infinity, is given by the distribution of [Formula: see text], where [Formula: see text] is a semicirlce random variable and [Formula: see text] is the [Formula: see text]th Chebychev polynomial.
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