On extremal spectral radius of blow-up uniform hypergraphs

Abstract

Let G be an r-uniform hypergraph of order t, and let ρ(G) be 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 vi1∈Vj1,vi2∈Vj2,…,vir∈Vjr. Let Bn(G) be the set of all the blow-ups of G such that each ni≥1 and ∑i=1tni=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 thatρ(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 ⌊n/t⌋ or ⌈n/t⌉. For any H∈Bn(SH(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(SH(m,q,r)), respectively.