Minimum Status, Matching and Domination of Graphs

Abstract

Given a graph, the status of a vertex is the sum of the distances between the vertex and all other vertices. The minimum status of a graph is the minimum of statuses of all vertices of this graph. We give a sharp upper bound for the minimum status of a connected graph with fixed order and matching number (domination number, respectively) and characterize the unique trees achieving the bound. We also determine the unique tree such that its minimum status is as small as possible when order and matching number (domination number, respectively) are fixed.