On a relationship between the characteristic and matching polynomials of a uniform hypertree

• Published in: Discrete mathematics Vol. 347; no. 5; p. 113915
• Main Authors: Li, Honghai, Su, Li, Fallat, Shaun
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.05.2024
• Subjects: Characteristic polynomial; Chip-firing game; Hypertree; Matching polynomial; Resultant; Tensors; Tensors; Chip-firing game; Resultant; Matching polynomial; Hypertree; Characteristic polynomial
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2024.113915

A hypertree is a connected hypergraph without cycles. Further a hypertree is called an r-tree if, additionally, it is r-uniform. Note that 2-trees are just ordinary trees. A classical result states that for any 2-tree T with characteristic polynomial ϕT(λ) and matching polynomial φT(λ), then ϕT(λ)=φT(λ). More generally, suppose T is an r-tree of size m with r≥2. In this paper, we extend the above classical relationship to r-trees and establish thatϕT(λ)=∏H⊑TφH(λ)aH, where the product is over all connected subgraphs H of T, and the exponent aH of the factor φH(λ) can be written asaH=bm−e(H)−|∂(H)|ce(H)(b−c)|∂(H)|, where e(H) is the size of H, ∂(H) is the boundary of H, and b=(r−1)r−1,c=rr−2. In particular, for r=2, the above correspondence reduces to the classical result for ordinary trees. In addition, we resolve a conjecture by Clark and Cooper (2018) [7] and show that for any subgraph H of an r-tree T with r≥3, φH(λ) divides ϕT(λ), and additionally ϕH(λ) divides ϕT(λ), if either r≥4 or H is connected when r=3. Moreover, a counterexample is given for the case when H is a disconnected subgraph of a 3-tree.