Geometric vs Algebraic Nullity for Hyperpaths

Cooper, Joshua M.; Fickes, Grant

doi:10.48550/arxiv.2107.01500

Cited by 1 publication

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“…This raises a natural question: Is it possible to determine the multiplicity of an eigenvalue of A(T ) as a root of the characteristic polynomial φ A(T ) (x)? For further discussion, see [15,8,6]. Clark and Cooper [3] conjectured the following: If T ′ is a subtree of a k-tree T with k ≥ 3, then ϕ(T ′ , x) divides φ A(T ) (x), where ϕ(T ′ , x) := r≥0 (−1) r p(T ′ , r)x (m(T ′ )−r)k .…”

Section: Discussionmentioning

confidence: 99%

Spectra of weighted uniform hypertrees

Wan¹,

Wang²,

Hu³

2022

Preprint

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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 bewhere 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 (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.

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