论文标题
加权均匀大体的光谱
Spectra of weighted uniform hypertrees
论文作者
论文摘要
令$ t $为$ k $ -tree，配备了加权功能$ \ w：v（t）\ cup e（t）\ rightarrow \ c $，其中$ k \ geq 3 $。加权匹配的$ k $ -tree $（t，\ w）$的加权匹配定义为$$μ（t，\ w，x）= \ sum _ {m \ in \ Mathcal {m} {m}（t）}（t）}（t）}（-1） \ prod_ {v \ in V（t）\ backslash v（m）}（x- \ w（v）），$$，其中$ \ mathcal {m}（m}（t）$表示$ t $的一组匹配项（包括空集）。在本文中，我们调查了加权$ k $ -tree $（t，\ w）$的邻接张量$ \ a（t，\ w）$的特征值。主要结果规定，$ \ w（v）$是v（t）$中每个$ v \的$ \ a（t，\ w）$的特征值，如果$ v（t）$ in v（t）$ in v（t）$中的每个$ v \ n neq \ neq \ w（v）$，则$λ$，则$λ$是$ \ w）of $ \ w） $ t $使$λ$是$μ（t'，\ w，x）$的根。此外，$ \ a（t，\ w）$的光谱半径等于$ \ w $是真实且不负的，$μ（t，\ w，x）$的最大根。结果将Clark and Cooper（{\ em在Hyperrees的邻接光谱上，Electron。J.Combin。，25（2）（2018）$ \＃$ P2.48}）扩展到加权$ k $ -trees。作为应用程序，获得了上述laplacian的两个类似物和$ k $树的无价的拉普拉斯张量。
Let $T$ be a $k$-tree equipped with a weighting function $\w: V(T)\cup E(T)\rightarrow \C$, where $k \geq 3$. The weighted matching polynomial of the weighted $k$-tree $(T,\w)$ is defined to be $$ μ(T,\w,x)= \sum_{M \in \mathcal{M}(T)}(-1)^{|M|}\prod_{e \in E(M)}\mathbf{w}(e)^k \prod_{v \in V(T)\backslash V(M)}(x-\w(v)), $$ where $\mathcal{M}(T)$ denotes the set of matchings (including empty set) of $T$. In this paper, we investigate the eigenvalues of the adjacency tensor $\A(T,\w)$ of the weighted $k$-tree $(T,\w)$. The main result provides that $\w(v)$ is an eigenvalue of $\A(T,\w)$ for every $v\in V(T)$, and if $λ\neq \w(v)$ for every $v\in V(T)$, then $λ$ is an eigenvalue of $\A(T,\w)$ if and only if there exists a subtree $T'$ of $T$ such that $λ$ is a root of $μ(T',\w,x)$. Moreover, the spectral radius of $\A(T,\w)$ is equal to the largest root of $μ(T,\w,x)$ when $\w$ is real and nonnegative. The result extends a work by Clark and Cooper ({\em On the adjacency spectra of hypertrees, Electron. J. Combin., 25 (2)(2018) $\#$P2.48}) to weighted $k$-trees. As applications, two analogues of the above work for the Laplacian and the signless Laplacian tensors of $k$-trees are obtained.
