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 $(n_{1}, n_{2},\ldots,n_{t})$, denoted by $G \circ (n_{1}, n_{2},\ldots,n_{t})$, 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 $\{j_{1},j_{2},\ldots,j_{r}\}\in E(G)$, then $\{v_{i_1},v_{i_2},\ldots,v_{i_r}\}\in E(H)$ for every $v_{i_{1}}\in V_{j_{1}}, v_{i_{2}}\in V_{j_{2}},\ldots, v_{i_{r}}\in V_{j_{r}}$. 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 $$\rho(K_{t}^{r}\circ(n-t+1,1,1,\ldots,1))\leq\rho(H)\leq \rho (T_{t}^{r}(n)),$$ with the left equality holds if and only if $H\cong K_{t}^{r}\circ(n-t+1,1,1,\ldots,1)$, 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.