On the walk matrix of the Dynkin graph Dn

Abstract

Let W(Dn) denote the walk matrix of the Dynkin graph Dn (n≥4), a tree obtained from the path of order n−1 by adding a pendant edge at the second vertex. We prove that rankW(Dn)=n−2 if 4|n and rankW(Dn)=n−1 otherwise. Furthermore, we prove that the Smith normal form of W(Dn) isdiag[1,1,…,1︸⌈n2⌉,2,2,…,2︸⌊n2⌋−1,0] when 4∤n. This confirms a recent conjecture in [W. Wang, F. Liu, W. Wang, Generalized spectral characterizations of almost controllable graphs, European J. Combin. 96(2021): 103348].