On the (reverse) cover cost of trees with some given parameters

Abstract

For a connected graph G=(VG,EG), the cover cost of a vertex u in G is defined as CCG(u)=∑v∈VGHuv and the reverse cover cost of a vertex v in G is defined as RCG(v)=∑u∈VGHuv, where Huv is the expected hitting time for random walk beginning at u to visit v. These two parameters were introduced as tractable variants of cover time on a graph. In this paper, some extremal problems on the (reverse) cover cost of a vertex in a tree with given graph parameters are considered. Firstly, the sharp lower bound on the cover cost among all n-vertex trees with given diameter (resp. matching number, domination number, vertex bipartition) is established. Secondly, the sharp lower bound on the reverse cover cost among all n-vertex trees with given diameter (resp. matching number, domination number) is established. All the corresponding extremal graphs are identified, respectively. At last, the unique n-vertex tree with given number of leaves having the minimum cover cost (resp. reverse cover cost) is characterized.