The effect of edge deletion on noncommutative distances on graphs

For a Dirac operator $D$ on a finite weighted graph, let $d^D$ denote the associated noncommutative (Connes) distance. We show that deleting an edge can actually decrease the noncommutative distance between two vertices in the graph. The smallest example of this phenomenon comes from a weighted 4-cycle, and the case of deleting an edge from a weighted 4-cycle is determined completely: if $d^{D^{\mathrm{\prime }}}$ denotes the noncommutative distance in the graph after edge deletion, we prove that $\sup d^D\mathrm{/}{d}^{D^{\mathrm{\prime }}}=2\mathrm{/}\sqrt{3}$ for the $4$-cycle, and further that this upper bound holds for all graphs. On the other hand, we show that deleting an edge can decrease the noncommutative distance between two vertices of an $n$-cycle precisely when $n$ is divisible by $4$. In cases when a deleted edge decreases the noncommutative distance between two vertices $x$ and $y$, we show that the deleted edge need not be incident to either of the vertices $x$ or $y$, and in fact the deleted edge can be arbitrarily far from both in the graph-theoretic (number of edges) distance and in the weighted (geodesic) distance. Moreover, deletion of a single vertex arbitrarily far from both $x$ and $y$ in either the graph-theoretic or weighted distance can change $d^D(x,y)$ by an arbitrarily large factor.