4 research outputs found
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 be an -uniform hypergraph of order and is the
spectral radius of , where is the adjacency
tensor of . A blow-up of respected to a positive integer vector , denoted by , is an
-uniform hypergraph obtained from by replacing each vertex of
with a class of vertices of size and if
, then
for every . Let
be the set of all the blow-ups of such that each
and . Let be the complete
-uniform hypergraph of order , and let be the -uniform
sunflower hypergraph with petals and a kernel of size on
vertices. For any , we prove that
with the left equality holds if and only if , and the right equality holds if and only
if , where is the complete -partite
-uniform hypergraph of order , with parts of size
or . For any , we
determine the exact value of the spectral radius of and characterize the
hypergraphs with maximum spectral radius and minimum spectral radius in
, 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