We investigate several measures of peripherality for vertices and edges in networks. We improve asymptotic bounds on the maximum value achieved by edge peripherality, edge sum peripherality, and the Trinajstić index over $n$ vertex graphs. We also prove similar results on the maxima over $n$-vertex bipartite graphs, trees, and graphs with a fixed diameter. Finally, we refute two conjectures of Furtula, the first on necessary conditions for minimizing the Trinajstić index and the second about maximizing the Trinajstić index.