2023/02/14 by Shao-Han Xu, Xu, Shao-Han, Futao Hu +3 · 1 citation
Mathematics · Neuroscience · #05C65 #15A18 #15A69 #Combinatorics (math.CO) #FOS: Mathematics #G.2.2 #Graph theory and applications #Nuclear Receptors and Signaling #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2302.06822
openalex publication_date 2023/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be an r-uniform hypergraph of order t and ρ(G) is the spectral radius of A(G), where A(G) is the adjacency tensor of G. A blow-up of G respected to a positive integer vector (n1, n2,…,nt), denoted by G ∘ (n1, n2,…,nt), is an r-uniform hypergraph obtained from G by replacing each vertex j of G with a class of vertices Vj of size nj≥ 1 and if \j1,j2,…,jr\∈ E(G), then \vi1,vi2,…,vir\∈ E(H) for every v_i1∈ V_j1, v_i2∈ V_j2,…, v_ir∈ V_jr. Let Bn(G) be the set of all the blow-ups of G such that each ni≥ 1 and ∑i=1n ni=n. Let Ktr be the complete r-uniform hypergraph of order t, and let SH(m,q,r) be the r-uniform sunflower hypergraph with m petals and a kernel of size r-q on t vertices. For any H∈ Bn(Ktr), we prove that ρ(Ktr∘(n-t+1,1,1,…,1))≤ρ(H)≤ ρ(Ttr(n)), with the left equality holds if and only if H≅ Ktr∘(n-t+1,1,1,…,1), and the right equality holds if and only if H≅ Ttr(n), where Ttr(n) is the complete t-partite r-uniform hypergraph of order n, with parts of size \lfloor n / k\rfloor or \lceil n / k \rceil. For any H∈ Bn(H(m,q,r)), we determine the exact value of the spectral radius of H and characterize the hypergraphs with maximum spectral radius and minimum spectral radius in Bn(H(m,q,r)), respectively.