Macroscopic scalar curvature bounds on graphs

We establish sharp lower bounds for the lengths of balls in the universal covers of finite metric graphs. For a finite connected $d$-regular graph, with $d \geq 3$ and average edge length at most one, we show that, at every radius, the average length of balls centered at a vertex of the universal cover is at least that of a ball of the same radius in the equilateral $d$-regular tree. We extend this comparison to arbitrary finite connected graphs whose total length does not exceed that of an equilateral trivalent graph with the same cyclomatic number. In particular, we show that the maximal length of a ball centered at a vertex of the universal cover is bounded below by that of a ball of the same radius in the equilateral trivalent tree. This answers a question of the first author. Furthermore, equality at an integer radius forces all edges to have unit length in the regular case and, in the general case, forces the graph to be an equilateral trivalent graph after suppressing vertices of degree two. In contrast, rigidity fails at every noninteger radius.