论文标题
光谱人造树
Spectral faux trees
论文作者
论文摘要
相对于给定矩阵的光谱人造树是一个不是树的图，而是具有给定矩阵的树的共光。我们考虑了几个矩阵的光谱人造树的存在，重点是建筑。 对于Laplacian矩阵，没有光谱人造树。对于邻接矩阵，几乎所有树木都是带有人造树的共同度。对于无标志性的拉普拉斯矩阵，只有在顶点的数量为$ n = 4k $时，光谱人造树才能存在。对于标准化的邻接，当顶点$ n \ ge 4 $的数量时，存在光谱人造树，我们为一个家庭提供了明确的结构，该家庭的尺寸以$ n =αk+1 $ $ n =α$的$ k $呈指数增长，其中$α$是固定的。
A spectral faux tree with respect to a given matrix is a graph which is not a tree but is cospectral with a tree for the given matrix. We consider the existence of spectral faux trees for several matrices, with emphasis on constructions. For the Laplacian matrix, there are no spectral faux trees. For the adjacency matrix, almost all trees are cospectral with a faux tree. For the signless Laplacian matrix, spectral faux trees can only exist when the number of vertices is of the form $n=4k$. For the normalized adjacency, spectral faux trees exist when the number of vertices $n\ge 4$, and we give an explicit construction for a family whose size grows exponentially with $k$ for $n=αk+1$ where $α$ is fixed.
