Abstract
Let \(\mathbb{S}\)(m, d, k) be the set of k-uniform supertrees with m edges and diameter d, and S1 (m, d, k) be the k-uniform supertree obtained from a loose path u1, e1, u2, e2, …, ud, ed, ud+1 with length d by attaching m–d edges at vertex u⌊d/2⌋+1. In this paper, we mainly determine S1 (m, d, k) with the largest signless Laplacian spectral radius in \(\mathbb{S}\)(m, d, k) for 3 ⩽ d ⩽ m − 1. We also determine the supertree with the second largest signless Laplacian spectral radius in \(\mathbb{S}\)(m, 3, k). Furthermore, we determine the unique k-uniform supertree with the largest signless Laplacian spectral radius among all k-uniform supertrees with n vertices and pendent edges (vertices).
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Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable suggestions to improve the presentation of this paper. This work was supported in part by the National Natural Science Foundation of China (Grant No. 11871398), the Natural Science Foundation of Shaanxi Province (Nos. 2020JQ-107, 2020JQ-696), and the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University (Nos. ZZ2018171, CX2020190).
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Duan, C., Wang, L. & Xiao, P. Largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges (vertices). Front. Math. China 15, 1105–1120 (2020). https://doi.org/10.1007/s11464-020-0879-0
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DOI: https://doi.org/10.1007/s11464-020-0879-0