On the minimum rank of the third power of a starlike tree

For a simple graph G of order n, let A be a real n×n symmetric matrix whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. The minimum rank of G is the smallest possible rank over all such symmetric matrices. The jth power of a graph G is the graph Gj=(V,F), where {u,v}∈F if and only if there is a walk of length j from u to v. In 2007 Brualdi, Hogben and Shader reported a conjecture that if T is not the star K1,n-1, then mr(T3)⩽mr(T2)-1. In this paper, we construct a class of starlike trees such that mr(T3)⩾mr(T2), which give a negative answer to this conjecture.