Chromatic polynomials of some sunflower mixed hypergraphs

The theory of mixed hypergraphs coloring has been first introduced by Voloshin in 1993 and it has been growing ever since. The proper coloring of a mixed hypergraph H = (X; C;D) is the coloring of the vertex set X so that no D-hyperedge is monochromatic and no C-hyperedge is polychromatic. A mixed hypergraph with hyperedges of type D, C or B is commonly known as a D-, C-, or B-hypergraph respectively, where B = C = D. D-hypergraph colorings are the classic hypergraph colorings which have been widely studied. The chromatic polynomial P(H;λ) of a mixed hypergraph H is the function that counts the number of proper λ-colorings, which are mappings. Recently, Walter published [15] some results concerning the chromatic polynomial of some non-uniform D-sunflower. In this paper, we present an alternative proof of his result and extend his formula to those of non-uniform C-sunflowers and B-sunflowers. Some results of a new but related member of sunflowers are also presented

On extremal spectral radius of blow-up uniform hypergraphs

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.Comment: 16 page

Chromatic polynomials of some sunflower mixed hypergraphs

The theory of mixed hypergraphs coloring has been first introduced by Voloshin in 1993 and it has been growing ever since. The proper coloring of a mixed hypergraph H = (X; C;D) is the coloring ofthe vertex set X so that no D??hyperedge is monochromatic and no C-hyperedge is polychromatic. A mixed hypergraph with hyperedges of type D, C or B is commonly known as a D-, C-, or B-hypergraphrespectively where B = C = D. D-hypergraph colorings are the classichypergraph colorings which have been widely studied. The chro-matic polynomial P(H;) of a mixed hypergraph H is the function thatcounts the number of proper ??colorings, which are mappings f : X !f1; 2; : : : ; g. A sunfower (hypergraph) with l petals and a core S is a collection of sets e1; : : : ; el such that ei \ ej = S for all i 6= j. Recently, Walter published [14] some results concerning the chromatic polynomial of some non-uniform D-sunfower. In this paper, we present an alternative proof of his result and extend his formula to those of non-uniform C-sunowers and B-sunowers. Some results of a new but related member of sunfowers are also presented