On the Metric Dimension of Biregular Graph

抄録

The metric dimension of a connected graph G is the minimum number of vertices in a subset W of V(G) such that all other vertices are uniquely determined by its vector distance to the vertices in W. In this paper, we consider a connected graph G where every vertex of G has relatively same probability to resolve some distinct vertices in G, namely a (μ, σ)-regular graph. We give tight lower and upper bounds on the metric dimension of a connected (μ, σ)-regular graphs of order n ≥ 2 where 1 ≤ µ ≤ n-1 and σ=n-1.