A Sharp Matching-Number Threshold for Spectral-Walk Determination of Trees

Abstract

The spectral characterization of graphs is a central problem in spectral graph theory. In this paper we study when a tree is determined, among trees, by its generalized spectrum. We use the equivalent formulation given by the adjacency spectrum together with the total-walk sequence $W_k(G)=1^{\mathsf{T}}A(G{)}^{k}1$. We determine the exact matching-number threshold for this tree-level reconstruction problem. If $T$ and $T^{\prime }$ are trees with matching number at most 4 and have the same adjacency spectrum and the same total-walk sequence, then $T\cong T^{\prime }$. Moreover, in this range it is enough to require equality of $W_k$ for $3\le k\le 8$. The bound is sharp: for every positive integer $m$ we construct a pair of non-isomorphic trees with matching number 5 having the same adjacency spectrum and identical total-walk sequences. The proof of the positive result is based on a finite-core reduction and an algebraic reconstruction of the possible pendant attachments.