The extremal graph of \(k\)th upper generalized exponents for a class of primitive graphs (Q2767410)

Graphs discussed in this paper allow loops, but not parallel edges. A graph \(G\) is called primitive if \(G\) is connected and \(G\) contains at least one odd cycle. (A loop is a cycle of length one.) Let \(X\) be a set of vertices of \(G\). Define \(\exp_G(X)\) to be the smallest positive integer \(p\) such that, for any vertex \(y\) of \(G\), there exists a path of length \(p\) from some vertex in \(X\) to \(y\). For a primitive graph \(G\) of order \(n\) and an integer \(k\), \(1\leq k\leq n-1\), let the \(k\)th upper generalized exponents be defined as \(F(G,k)= \max\{\exp_G(X)\mid X\subset V(G), |X|= k\}\). For \(1\leq d\leq n\) and \(2\leq k\leq n-1\), let \(F(n,d,k)\) denote the largest \(k\)th upper generalized exponent of a primitive graph of order \(n\) with \(d\) loops. The values of \(F(n,d,k)\) have already been completely determined. This paper characterizes all extremal graphs that attain the maximum exponents \(F(n,d,k)\).