Let G be a labeled, connected, bipartite graph with the bi-partition $(L=\{l_1,\dots ,l_k\},R=\{r_1,\dots ,r_p\})$ of the vertex set V. Let D be the usual distance matrix of G, where rows and columns are indexed by $l_1,\dots ,l_k,r_1,\dots ,r_p$. For $X,Y\subseteq V$, let us define $D_G[X,Y]$ as the submatrix of D induced by the rows indexed in X and columns indexed in Y. Let us call $D_G[L,R]$ the bipartite distance matrix of G. If G has a unique perfect matching, then $k=p$ and we assume that the bi-partition is canonical, that is, $[l_i,r_i]$ are matching edges. For a nonsingular tree T, let us denote the bipartite distance matrix of T by $\mathcal{B}(T)$.

We observe that $\det \mathcal{B}(T)$ is always a multiple of $2^{p-1}$. This is similar to the well known result of Graham and Pollak (1971) [1] which tells that the determinant of the usual distance matrix D is a multiple of $2^{n-2}$. Call the number $\mathtt{bd}(T):=\det \mathcal{B}(T)/{(-2)}^{p-1}$ the bipartite distance index of T. We supply a recursive formula to compute this index. We show that this index satisfies an interesting inclusion-exclusion type of principle at any matching edge of the tree. Even more interestingly, we show that the index is completely characterized by the structure of T via what we call the f-alternating sums, that is, the sum $f(T):=\sum [d(u)-2][d(v)-2]S_{|P_{uv}|/2}$, where the sum is taken over all u-v-alternating paths $P_{uv}$, and S is the sequence $(1,1,3,3,5,5,\dots )$.

A well known result by Graham, Hoffman and Hosoya (1977) [2] is that the determinant of the distance matrix of a graph only depends on the blocks and is independent of how they are assembled. Such a result does not hold true for $\mathcal{B}(T)$. However, we identify some basic elements and a merging operation and show that each of the trees that can be constructed from a given set of elements, sequentially using this operation, have the same $\det \mathcal{B}(T)$, independent of the order in which the sequence is followed. For the class of trees that can be obtained in this way, we give a surprisingly simple way to evaluate the determinant of $\mathcal{B}(T)$.

令G是一个带标签的、连通的、二分图$(升=\{{升}_1,\dots ,{升}_{克}\},\mathrm{电阻}=\{r_1,\dots ,r_{磷}\})$的顶点集V。令D为G的常用距离矩阵，其中行和列的索引为${升}_1,\dots ,{升}_{克},r_1,\dots ,r_{磷}$. 为了$X,是\subseteq 伏$，让我们定义 $D_G[X,是]$作为D的子矩阵，由X 中索引的行和Y 中索引的列引起。让我们打电话$D_G[升,\mathrm{电阻}]$G的二部距离矩阵。如果G有唯一的完美匹配，则$克=磷$ 我们假设双分区是规范的，也就是说， $[{升}_{\mathrm{一世}},r_{\mathrm{一世}}]$是匹配边。对于非奇异树牛逼，让我们代表的二部距离矩阵牛逼通过$\mathcal{乙}(吨)$.

我们观察到 $\mathrm{检测}\mathcal{乙}(吨)$ 始终是的倍数 $2^{磷-1}$. 这类似于 Graham 和 Pollak (1971) [1] 的众所周知的结果，该结果表明通常距离矩阵D 的行列式是$2^{n-2}$. 拨打号码$\mathtt{BD}(吨)：=\mathrm{检测}\mathcal{乙}(吨)/{(-2)}^{磷-1}$的二分距离指数的Ť。我们提供了一个递归公式来计算这个指数。我们表明，该索引在树的任何匹配边上都满足有趣的包含-排除类型的原则。更有趣的是，我们表明，该指数是完全特征的结构牛逼通过我们所说的F-交替的资金，也就是总和$F(吨)：=\sum [d(你)-2][d(v)-2]{秒}_{|{磷}_{你v}|/2}$，其中所有u - v交替路径的总和${磷}_{你v}$, S是序列$(1,1,3,3,5,5,\dots )$.

Graham、Hoffman 和 Hosoya (1977) [2] 的一个众所周知的结果是图的距离矩阵的行列式仅取决于块，而与它们的组装方式无关。这样的结果并不适用于$\mathcal{乙}(吨)$. 但是，我们确定了一些基本元素和合并操作，并表明可以从给定元素集构造的每个树，依次使用此操作，具有相同的$\mathrm{检测}\mathcal{乙}(吨)$, 与序列遵循的顺序无关。对于可以通过这种方式获得的树类，我们给出了一个令人惊讶的简单方法来评估$\mathcal{乙}(吨)$.