论文标题
学位距离和传输矩阵
The degree-distance and transmission-adjacency matrices
论文作者
论文摘要
令$ g $为具有邻接矩阵$ a（g）$的连接图。 $ g $的距离矩阵$ d（g）$具有$ v（g）$索引的行和列，$ uv $ -entry等于$ \ mathrm {dist}（u，v）$，这是边缘$ u $和$ u $和$ v $之间最短路径中的边缘数。 $ u $的传输$ \ mathrm {trs}（u）$定义为$ \ sum_ {v \ in V（g）} \ Mathrm {dist}（u，v）$。令$ \ mathrm {trs}（g）$为对角线的对角矩阵，对角的$ g $的传输和$ \ mathrm {deg}（g）$对角矩阵具有对角度的矩阵。在本文中，我们调查了史密斯的正常形式（SNF）和矩阵的光谱$ d^{\ mathrm {deg}} _+（g）：= \ m缩美元特别是，我们探讨了这些矩阵的频谱和SNF的良好，以确定图表到同构。我们发现，与其他经典矩阵相比，$ a^{\ mathrm {trs}} $的SNF具有有趣的行为。我们注意到，可以使用$ a^{\ mathrm {trs}} $的SNF来计算某些图的沙皮群的结构。我们计算$ d^{\ mathrm {deg}} _+$，$ d^{\ mathrm {deg}} $，$ d^{\ mathrm {deg}} $，$ a^{\ mathrm {trs}} _+$和$ a^{\ Mathrm {TRS}} $的snf。我们证明，完整的图是由$ d^{\ Mathrm {deg}} _+$，$ d^{\ Mathrm {deg}} $，$ a^{\ Mathrm {trs}} _+$和$ a^and $ a^{\ mathrm {\ Mathrm {trs {trs {最后，我们得出了有关$ d^{\ mathrm {deg}} $和$ a^{\ mathrm {trs}} $的频谱的一些结果。
Let $G$ be a connected graph with adjacency matrix $A(G)$. The distance matrix $D(G)$ of $G$ has rows and columns indexed by $V(G)$ with $uv$-entry equal to the distance $\mathrm{dist}(u,v)$ which is the number of edges in a shortest path between the vertices $u$ and $v$. The transmission $\mathrm{trs}(u)$ of $u$ is defined as $\sum_{v\in V(G)}\mathrm{dist}(u,v)$. Let $\mathrm{trs}(G)$ be the diagonal matrix with the transmissions of the vertices of $G$ in the diagonal, and $\mathrm{deg}(G)$ the diagonal matrix with the degrees of the vertices in the diagonal. In this paper we investigate the Smith normal form (SNF) and the spectrum of the matrices $D^{\mathrm{deg}}_+(G):=\mathrm{deg}(G)+D(G)$, $D^{\mathrm{deg}}(G):=\mathrm{deg}(G)-D(G)$, $A^{\mathrm{trs}}_+(G):=\mathrm{trs}(G)+A(G)$ and $A^{\mathrm{trs}}(G):=\mathrm{trs}(G)-A(G)$. In particular, we explore how good the spectrum and the SNF of these matrices are for determining graphs up to isomorphism. We found that the SNF of $A^{\mathrm{trs}}$ has an interesting behaviour when compared with other classical matrices. We note that the SNF of $A^{\mathrm{trs}}$ can be used to compute the structure of the sandpile group of certain graphs. We compute the SNF of $D^{\mathrm{deg}}_+$, $D^{\mathrm{deg}}$, $A^{\mathrm{trs}}_+$ and $A^{\mathrm{trs}}$ for several graph families. We prove that complete graphs are determined by the SNF of $D^{\mathrm{deg}}_+$, $D^{\mathrm{deg}}$, $A^{\mathrm{trs}}_+$ and $A^{\mathrm{trs}}$. Finally, we derive some results about the spectrum of $D^{\mathrm{deg}}$ and $A^{\mathrm{trs}}$.
