A complete solution of the k-uniform supertrees with the eight largest α-spectral radii

2023/08/01 by Yu, Lou-Jun, Wang, Wen-Huan
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.00422

Abstract

Let \mathcal T (n, k) be the set of the k-uniform supertrees with n vertices and m edges, where k≥ 3, n≥ 5 and m=(n-1)/(k-1). % Let m be the number of the edges of the supertrees in \mathcal T (n, k), where m=(n-1)/(k-1). A conjecture concerning the supertrees with the fourth through the eighth largest α-spectral radii in \mathcal T (n, k) was proposed by You et al. (2020), where 0 ≤ α<1, k≥ 3 and m ≥ 10. This conjecture was partially solved for 1-(1)/(m-2)≤ α<1 and m≥ 10 by Wang et al. (2022). When 0≤ α<1-(1)/(m-2) and m ≥ 10, whether this conjecture is correct or not remains a problem to be further solved. By using a new ρ_α-normal labeling method proposed in this article for computing the α-spectral radius of the k-uniform hypergraphs, we completely prove that this conjecture is right for 0≤α<1 and m≥ 13.

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