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Theκ-point Exponent Set of Central Symmetric Primitive Matrices

Let G=(V,E)be a primitive digraph.The vertex exponent of G at a vertex v∈V,denoted by exp_G(v),is the least integer p such that there is a v→u walk of length p for each u∈V.We choose to order the vertices of G in such a way that exp_G(v_1)≤exp_G(v_2)≤…≤exp_G(v_n).Then exp_G(v_k)is called the k-point exponent of G and is denoted by exp_G(k),1≤k≤n.We define the k-point exponent set E(n,k):={exp_G(k)|G=G(A)with A∈CSP(n)},where CSP(n)is the set of all n×n central symmetric primitive matrices and G(A)is the associated graph of the matrix A.In this paper,we describe E(n,k)for all n,k with 1≤k≤n except n≡1(mod 2)and 1≤k≤n-4.We also characterize the extremal graphs when k=1.