论文标题
离散的Bakry-émery曲率张量和连接图的矩阵
Discrete Bakry-Émery curvature tensors and matrices of connection graphs
论文作者
论文摘要
连接图是Harary签名图的自然扩展。 Liu，Münch和Peyerimhoff引入了连接图的Bakry-émery曲率，以建立用于连接Laplacians的Buser Type Eigenvalue估计值。在本文中，我们根据一个单位等效曲率矩阵的家族的最小特征值重新制定了连接图中顶点的Bakry-émery曲率。我们进一步将这个曲率矩阵家族解释为新定义的曲率张量的矩阵表示，相对于顶点处的切线空间的不同正顺序基础。这是库欣 - kamtue-liu-peyerimhoff和siconolfi在图形曲率矩阵上的先前作品的强烈扩展。此外，我们研究了连接图的笛卡尔产品的Bakry-émery曲率，从而增强了Liu，Münch和Peyerimhoff的先前结果。尽管具有局部平衡结构的顶点的结果涵盖了先前的作品，但局部不平衡连接结构的各种有趣现象已得到阐明。
Connection graphs are natural extensions of Harary's signed graphs. The Bakry-Émery curvature of connection graphs has been introduced by Liu, Münch and Peyerimhoff in order to establish Buser type eigenvalue estimates for connection Laplacians. In this paper, we reformulate the Bakry-Émery curvature of a vertex in a connection graph in terms of the smallest eigenvalue of a family of unitarily equivalent curvature matrices. We further interpret this family of curvature matrices as the matrix representations of a new defined curvature tensor with respect to different orthonormal basis of the tangent space at a vertex. This is a strong extension of previous works of Cushing-Kamtue-Liu-Peyerimhoff and Siconolfi on curvature matrices of graphs. Moreover, we study the Bakry-Émery curvature of Cartesian products of connection graphs, strengthening the previous result of Liu, Münch and Peyerimhoff. While results of a vertex with locally balanced structure cover previous works, various interesting phenomena of locally unbalanced connection structure have been clarified.
