Further results on the expected hitting time, the cover cost and the related invariants of graphs

TL;DR

This paper explores the relationships between hitting times, Kirchhoff index, and resistance-centrality in unicyclic graphs, providing sharp bounds on cover costs and identifying extremal graphs using graph transformations.

Contribution

It establishes new bounds on cover costs in unicyclic graphs and identifies extremal structures, advancing understanding of graph invariants related to random walks.

Findings

Sharp upper and lower bounds on cover cost and reverse cover cost.

Identification of extremal unicyclic graphs for these bounds.

Demonstration of relations between hitting times, Kirchhoff index, and resistance-centrality.

Abstract

A close relation between hitting times of the simple random walk on a graph, the Kirchhoff index, resistance-centrality, and related invariants of unicyclic graphs is displayed. Combining with the graph transformations and some other techniques, sharp upper and lower bounds on the cover cost (resp. reverse cover cost) of a vertex in an $n$-vertex unicyclic graph are determined. All the corresponding extremal graphs are identified, respectively.