论文标题
具有基质重量的树：拉普拉斯矩阵和特征性顶点
Trees with Matrix Weights: Laplacian Matrix and Characteristic-like Vertices
论文作者
论文摘要
众所周知，通过Perron值和Perron分支在边缘上有正权重的树的特征顶点有一个替代表征。此外，具有正边缘重量的树的代数连通性可以用Perron值表示。 在本文中，我们考虑了矩阵权重的树木的边缘。更确切地说，我们对具有以下矩阵边缘权重的树木感兴趣：1。阳性确定的矩阵重量，2。下（或上）三角矩阵重量，带正对角线呈正。对于具有上述矩阵边缘权重的树的树，我们定义了Perron值和Perron分支。此外，我们已经展示了满足特性的顶点的存在，类似于在Perron值和Perron分支方面具有正边缘权重的树的特性，我们称此类顶点具有特征性的类似的顶点。在这种情况下，拉普拉斯矩阵的特征值是非负的，我们在laplacian矩阵的第一个非零特征值中获得了下限，就perron值而言。此外，我们还计算了一棵树的拉普拉斯矩阵的摩尔 - 芬罗倒数，其边缘上有非矩阵重量。
It is known that there is an alternative characterization of characteristic vertices for trees with positive weights on their edges via Perron values and Perron branches. Moreover, the algebraic connectivity of a tree with positive edge weights can be expressed in terms of Perron value. In this article, we consider trees with matrix weights on their edges. More precisely, we are interested in trees with the following classes of matrix edge weights: 1. positive definite matrix weights, 2. lower (or upper) triangular matrix weights with positive diagonal entries. For trees with the above classes of matrix edge weights, we define Perron values and Perron branches. Further, we have shown the existence of vertices satisfying properties analogous to the properties of characteristic vertices of trees with positive edge weights in terms of Perron values and Perron branches, and we call such vertices characteristic-like vertices. In this case, the eigenvalues of the Laplacian matrix are nonnegative, and we obtain a lower bound for the first non-zero eigenvalue of the Laplacian matrix in terms of Perron value. Furthermore, we also compute the Moore-Penrose inverse of the Laplacian matrix of a tree with nonsingular matrix weights on its edges.
