Generalized indices of non-primitive graphs (Q1865991)

Suppose \(G\) is a digraph and \((x,y)\) is any ordered pair of vertices in \(G\). Let \(k\) be the minimum nonnegative integer and \(p\) the minimum positive integer corresponding to \(G\) such that there is a walk of length \(k\) from \(x\) to \(y\) if and only if there is a walk of length \(k+p\) from \(x\) to \(y\). The index and period of \(G\) are \(k\) and \(p\), respectively. The author determines the maximum value of the generalized index of a class of non-primitive graphs of order \(n\) and of a class of non-primitive simple graphs of order \(n\). In addition, the generalized index sets for a class of bipartite graphs of order \(n\) are determined.