Block eccentricity, a radius bound, and an application to the Randi\'c index

We propose a framework for thinking about eccentricity in terms of blocks. We extend the familiar definitions of radius and center to blocks and verify that a central block contains all central points. We classify graphs into two types depending upon the relationship between block radius and vertex radius and between central blocks and central vertices; from this we derive a new lower bound on diameter in terms of the diameter of the central block. We also identify a subgraph which respects the block structure of the original graph and realizes the same vertex radius, and we use it to verify that cactus graphs satisfy a conjectured bound between vertex radius and the Randic index, an invariant from mathematical chemistry.