Some further results on the maximal hitting times of trees with some given parameters

Abstract

Given a connected graph G with vertex set VG, the hitting time HG(u,v) of vertices u and v in G is defined as the expected number of steps that a random walk takes to go from u to v. Then the ZZ index of G, denoted by ψ(G), is defined as ψ(G)=max{u,v}⊆VGHG(u,v). This hitting-time-based invariant was first proposed by Zhu and Zhang (2021). In this paper, some further extremal problems on the ZZ index of trees with some given parameters are considered. Firstly, the extremal graphs with the second largest and the third largest ZZ indices are characterized among all the trees with given diameter (resp. matching number, domination number, vertex bipartition, the number of leaves). Secondly, the unique graph with the largest ZZ index is determined among all the trees with given segment sequence (resp. the number of segments).