2022/03/29 by Jiang-Chao Wan, Yi Wang, Wan, Jiang-Chao +3
Mathematics · Medicine · #Combinatorics #Eigenvalues and eigenvectors #Graph theory and applications #Lambda #Laplace operator #Mathematics #Physics #Phytoestrogen effects and research #Quantum mechanics #Tensor decomposition and applications #Tree (set theory)
paper · pdf · doi:10.48550/arxiv.2203.15339
openalex publication_date 2022/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let T be a k-tree equipped with a weighting function \w: V(T)∪ E(T)→ \C, where k ≥ 3. The weighted matching polynomial of the weighted k-tree (T,\w) is defined to be μ(T,\w,x)= ∑M ∈ M(T)(-1)|M|∏e ∈ E(M)w(e)k ∏v ∈ V(T)\backslash V(M)(x-\w(v)), where 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∈ V(T), and if λ≠ \w(v) for every v∈ 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.