当前位置: X-MOL 学术Linear Algebra its Appl. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
The bipartite distance matrix of a nonsingular tree
Linear Algebra and its Applications ( IF 1.0 ) Pub Date : 2021-09-06 , DOI: 10.1016/j.laa.2021.09.001
R.B. Bapat 1 , Rakesh Jana 2 , S. Pati 2
Affiliation  

Let G be a labeled, connected, bipartite graph with the bi-partition (L={l1,,lk},R={r1,,rp}) of the vertex set V. Let D be the usual distance matrix of G, where rows and columns are indexed by l1,,lk,r1,,rp. For X,YV, let us define DG[X,Y] as the submatrix of D induced by the rows indexed in X and columns indexed in Y. Let us call DG[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, [li,ri] are matching edges. For a nonsingular tree T, let us denote the bipartite distance matrix of T by B(T).

We observe that detB(T) is always a multiple of 2p1. 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 2n2. Call the number bd(T):=detB(T)/(2)p1 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):=[d(u)2][d(v)2]S|Puv|/2, where the sum is taken over all u-v-alternating paths Puv, and S is the sequence (1,1,3,3,5,5,).

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 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 detB(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 B(T).



中文翻译:

非奇异树的二部距离矩阵

G是一个带标签的、连通的、二分图(={1,,},电阻={r1,,r})的顶点集V。令DG的常用距离矩阵,其中行和列的索引为1,,,r1,,r. 为了X,,让我们定义 DG[X,]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DG[,电阻]G的二部距离矩阵。如果G有唯一的完美匹配,则= 我们假设双分区是规范的,也就是说, [一世,r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().

我们观察到 检测() 始终是的倍数 2-1. 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2n-2. 拨打号码BD()=检测()/(-2)-1二分距离指数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()=[d()-2][d(v)-2]|v|/2,其中所有u - v交替路径的总和v, S是序列(1,1,3,3,5,5,).

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(). 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,具有相同的检测(), 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().

更新日期:2021-09-15
down
wechat
bug