Extremal bounds on peripherality measures

Several measures of peripherality for vertices and edges in networks are investigated. Asymptotic bounds are improved on the maximum value achieved by the (i) edge peripherality over connected n-vertex graphs, (ii) edge sum peripherality over connected n-vertex graphs, (iii) edge sum peripherality over n-vertex graphs with diameter at most 2, (iv) edge sum peripherality over bipartite n-vertex graphs with diameter at most 3, and (v) Trinajstic index over n-vertex graphs. The maximum value achieved by the peripherality index over connected n-vertex graphs and n-vertex trees is also computed for all 1 <= n <= 8. Two conjectures of Furtula are refuted; the first one is on necessary conditions for minimizing the Trinajstic index and the second one is about maximizing the Trinajstic index. Finally, an asymptotic expression for the expected value of the irregularity of the random graph G(n,p) is found for arbitrary p satisfying 0 < p < 1.