Yang Yang, Ke Wei, Wang Zhe, Qiao Haiyan
College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China.
College of Artificial Intelligence, Tianjin University of Science and Technology, Tianjin 300457, China.
Heliyon. 2024 Jun 4;10(12):e32235. doi: 10.1016/j.heliyon.2024.e32235. eCollection 2024 Jun 30.
Consider a simple undirected connected graph , with and representing its degree and adjacency matrices, respectively. Furthermore, is the Laplacian matrix of , and is the heat kernel (HK) of , with denoting the time variable. For a vertex , the th element of the diagonal of the HK is defined as , and is the HK trace of , where denote the eigenvalues of . This study provides new computational formulas for the HK diagonal entries of graphs using an almost equitable partition and the Schur complement technique. We also provide bounds for the HK trace of the graphs.
考虑一个简单的无向连通图，分别用 (D) 和 (A) 表示其度矩阵和邻接矩阵。此外，(L) 是该图的拉普拉斯矩阵，(H_t) 是该图的热核（HK），其中 (t) 表示时间变量。对于一个顶点 (v)，热核对角线上的第 (k) 个元素定义为 (H_{kk}(v,t))，(\text{Tr}(H_t)) 是热核的迹，其中 (\lambda_i) 表示 (L) 的特征值。本研究使用几乎公平划分和舒尔补技术为图的热核对角元素提供了新的计算公式。我们还给出了图的热核迹的界。
