We study the free product of rooted graphs and its various decompositions using quantum probabilistic methods. We show that the free product of rooted graphs is canonically associated with free independence, which completes the proof of the conjecture that there exists a product of rooted graphs canonically associated with each notion of noncommutative independence which arises in the axiomatic theory. Using the `orthogonal product' of rooted graphs, we decompose the branches of the free product of rooted graphs as `alternating orthogonal products'. This leads to alternating decompositions of the free product itself, with the star product or the comb product followed by orthogonal products. These decompositions correspond to the recently studied decompositions of the free additive convolution of probability measures in terms boolean and orthogonal convolutions, or monotone and orthogonal convolutions. We also introduce a new type of `quantum decomposition' of the free product of graphs, where the distance partition of the set of vertices is taken with respect to a set of vertices instead of a single vertex. We show that even in the case of widely studied graphs this yields new and more complete information on their spectral properties, like spectral measures of a (usually infinite) set of cyclic vectors under the action of the adjacency matrix.
Gromov Product Decomposition of 7-point Metric Spaces
设$X$是具有元素$P_i$，$i=1,\dots,n$和距离函数$d_{ij}$的有限度量空间.三角形与顶点$P_i$、$P_j$和$P_k$在顶点$P_i$处的格罗莫夫积由$Δ_{ijk}=\frac{1}{2}(d_{ij}+d_{ik}-d_{jk})$定义。度量空间称为$Δ$ -通用的，如果在每个$P_i$上的Gromov乘积的集合具有唯一的最小元素$Δ_{ijk}$。对于$Δ$ -通用度量空间，映射$P_i\to (P_jP_k)$（其中$(P_jP_k)$是将$P_j$连接到$P_k$的边）是被称为“Gromov乘积结构”的良好定义的映射[Bilge，Celik和Kocak，“An equivalence class decomposition of finite度量空间“，Discrete Mathmetics，1930，（2017）1928-1932]。对于n=5，3个$Δ$等价类与$5$点度量的分类一致.对于$n=6$，有26个$Δ$等价类，由339个度量类细化。本文首次对7点空间进行了系统的处理，得到了由431个等价类组成的7点度量空间的PDMATHAMT-等价分解。
图$G$的分解是$G$的一族子图，其边集形成$E(G)$的一个划分。本文证明了每一个无三角形可平面图$G$都可以分解为一个$2$ -退化图和一个匹配。因此，每个无三角形可平面图$G$都有一个匹配的$M$，使得$G-M$是在线3-DP可着色的。这加强了[R.Škrekovski，{ \em A Grötzsch-Type Theorem for List Colorings with Impropriety One}，Combin.概率计算8（1999），493-507]证明了每个无三角形可平面图都是$1$ -亏$3$ -可选的。
