The kkth local exponent of doubly symmetric primitive matrices

Let D=(V,E)D=(V,E) be a primitive digraph. The local exponent of DD at a vertex u∈Vu∈V, denoted by γD(u)γD(u), is the least integer pp such that there is an u→vu→v walk of length pp for each v∈Vv∈V. Let V={v1,v2,…,vn}V={v1,v2,…,vn}. Following Brualdi and Liu, we order the vertices of VV so that γD(v1)≤γD(v2)≤⋯≤γD(vn)γD(v1)≤γD(v2)≤⋯≤γD(vn). Then γD(vk)γD(vk) is called the kkth local exponent of DD and is denoted by expD(k)expD(k), 1≤k≤n1≤k≤n. In this work we define exp(n,k)=max{expG(k)|G=G(A) with A∈DSP(n)}, where DSP(n) is the set of all n×nn×n doubly symmetric primitive matrices and G(A)G(A) is the associated graph of matrix AA. For n≥3n≥3, we determine that exp(n,k)=n−1exp(n,k)=n−1 for all kk with 1≤k≤n1≤k≤n.