一类度距离矩阵的同时同谱树族
A Family of Simultaneously Cospectral Trees for Degree-Distance Matrices
AI总结：
本文通过构造无穷多对顶点数为17r-15的非同构树，否定了所有树由Ddegp和Ddeg的谱唯一确定的猜想，该构造基于r正则化叶子扩展和等价划分约简。
AI中文摘要：
各类图矩阵的图谱刻画是谱图论的核心研究主题。设G为图，其邻接矩阵为A(G)，对角度矩阵为Deg(G)，距离矩阵为D(G)，传输矩阵为Trs(G)。近期Alfaro与Zapata（2024）引入度距离矩阵Ddegp(G)=Deg(G)+D(G)、Ddeg(G)=Deg(G)-D(G)，以及传输邻接矩阵Atrsp(G)=Trs(G)+A(G)、Atrs(G)=Trs(G)-A(G)。基于对顶点数不超过20的树的计算证据，他们猜想所有树都由Ddegp和Ddeg的谱唯一确定。本文通过构造无穷多对非同构树来否定这些猜想。具体而言，对每个整数r≥3，本文构造一对顶点数为17r-15的树，它们关于以下六个矩阵同时同谱：A、L、Q、D、Ddegp、Ddeg。该构造基于r正则化叶子扩展和等价划分约简。本文还记录了传输邻接矩阵的一个简单符号切换性质：若G是二分图，则Atrs(G)与Atrsp(G)通过对角{±1}矩阵相似，且具有相同的史密斯标准形。因此，对于树而言，Atrs和Atrsp的谱问题与史密斯标准形问题等价。
英文摘要：
Spectral characterization of graphs for various graph matrices constitutes a central topic in spectral graph theory. Let $G$ be a graph with adjacency matrix $A(G)$, diagonal degree matrix $\Deg(G)$, distance matrix $D(G)$, and transmission matrix \(\Trs(G)\), respectively. Recently, Alfaro and Zapata (2024) introduced the degree-distance matrices \(\Ddegp(G)=\Deg(G)+D(G)\) and \(\Ddeg(G)=\Deg(G)-D(G)\), together with the transmission-adjacency matrices \(\Atrsp(G)=\Trs(G)+A(G)\) and \(\Atrs(G)=\Trs(G)-A(G)\). Based on computational evidence for trees on at most \(20\) vertices, they conjectured that all trees are determined by the spectra of \(\Ddegp\) as well as \(\Ddeg\). In this paper, we disprove these conjectures by constructing an infinite family of pairs of non-isomorphic trees. More precisely, for each integer \(r\ge 3\), we construct a pair of trees on \(17r-15\) vertices which are simultaneously cospectral with respect to the following six matrices \[ A,\quad L,\quad Q,\quad D,\quad \Ddegp,\quad \Ddeg . \] The construction is based on an \(r\)-regularized leaf extension and an equitable-partition reduction. We also record a simple sign-switching observation for transmission-adjacency matrices: if \(G\) is bipartite, then \(\Atrs(G)\) and \(\Atrsp(G)\) are similar via a diagonal \(\{\pm1\}\)-matrix and have the same Smith normal form. Consequently, for trees, the spectral and Smith normal form problems for \(\Atrs\) and \(\Atrsp\) are equivalent.
