Extremal problems on k-ary trees with respect to the cover cost and reverse cover cost

Abstract

Let Tn,k be the set of k-ary trees of order n=(k−1)t+2, where k-ary trees are trees in which every vertex has degree 1 or k. For a connected graph G=(V(G),E(G)), the cover cost (resp. reverse cover cost) of a vertex u in G is defined as CCG(u)=∑v∈V(G)Huv (resp. RCG(u)=∑v∈V(G)Hvu), where Huv is the expected hitting time for random walk beginning at u to visit v. In this paper, we consider extremal problems on k-ary trees with respect to the cover cost and reverse cover cost. In particular, the maximum (resp. minimum) cover cost and reverse cover cost among all k-ary trees with given order are identified.