On extremal spectral radius of blow-up uniform hypergraphs

Abstract

Let $G$ be an $r$-uniform hypergraph of order $t$ and $\rho (G)$ is the spectral radius of $\mathcal{A}(G)$, where $\mathcal{A}(G)$ is the adjacency tensor of $G$. A blow-up of $G$ respected to a positive integer vector $(n_1,n_2,\dots ,n_t)$, denoted by $G\circ (n_1,n_2,\dots ,n_t)$, is an $r$-uniform hypergraph obtained from $G$ by replacing each vertex $j$ of $G$ with a class of vertices $V_j$ of size $n_j\ge 1$ and if $\{j_1,j_2,\dots ,j_r\}\in E(G)$, then $\{v_{i_1},v_{i_2},\dots ,v_{i_r}\}\in E(H)$ for every $v_{i_1}\in V_{j_1},v_{i_2}\in V_{j_2},\dots ,v_{i_r}\in V_{j_r}$. Let ${\mathcal{B}}_n(G)$ be the set of all the blow-ups of $G$ such that each $n_i\ge 1$ and ${\sum }_{i=1}^n{n}_i=n$. Let $K_t^r$ 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 $H\in {\mathcal{B}}_n(K_t^r)$, we prove that $$\rho (K_t^r\circ (n-t+1,1,1,\dots ,1))\le \rho (H)\le \rho (T_t^r(n)),$$ with the left equality holds if and only if $H\cong K_t^r\circ (n-t+1,1,1,\dots ,1)$, and the right equality holds if and only if $H\cong T_t^r(n)$, where $T_t^r(n)$ 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 $H\in {\mathcal{B}}_n(H(m,q,r))$, we determine the exact value of the spectral radius of $H$ and characterize the hypergraphs with maximum spectral radius and minimum spectral radius in ${\mathcal{B}}_n(H(m,q,r))$, respectively.

16 pages