Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSigned a-polynomials of graphs and Poincare polynomials of real toric manifolds
150
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Recently, Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. We introduce a signed a-polynomial which is a generalization of the signed a-number and gives a-, b-, and c-numbers. The signed a-polynomial of a graph $G$ is related to the Poincare polynomial $P_{M(G)}(z)$, which is the generating function for the Betti numbers of the real toric manifold $M(G)$. We give the generating functions for the signed a-polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. As a consequence, we find the Euler characteristic number and the Betti numbers of the real toric manifold $M(G)$ for complete multipartite graphs $G$.
rate research
Read More
In this paper, we study Eulerian polynomials for permutations and signed permutations of the multiset ${1,1,2,2,ldots,n,n}$. Properties of these polynomials, including recurrence relations and unimodality are discussed. In particular, we give a unified proof of the fact that these polynomials are unimodal with modes in the middle.
In recent joint work (2021), we introduced a novel multivariate polynomial attached to every metric space - in particular, to every finite simple connected graph $G$ - and showed it has several attractive properties. First, it is multi-affine and real-stable (leading to a hitherto unstudied delta-matroid for each graph $G$). Second, the polynomial specializes to (a transform of) the characteristic polynomial $chi_{D_G}$ of the distance matrix $D_G$; as well as recovers the entire graph, where $chi_{D_G}$ cannot do so. Third, the polynomial encodes the determinants of a family of graphs formed from $G$, called the blowups of $G$. In this short note, we exhibit the applicability of these tools and techniques to other graph-matrices and their characteristic polynomials. As a particular case, we will see that the adjacency characteristic polynomial $chi_{A_G}$ is in fact the shadow of a richer multivariate blowup-polynomial, which is similarly multi-affine and real-stable. Moreover, this polynomial encodes not only the aforementioned three properties, but also yields additional information for specific families of graphs.
We propose and prove an identity relating the Poincare polynomials of stabilizer subgroups of the affine Weyl group and of the corresponding stabilizer subgroups of the Weyl group.
In this paper, we present some properties on chromatic polynomials of hypergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain dense real zeros in the set of real numbers. We then prove that for any multigraph $G=(V,E)$, the number of totally cyclic orientations of $G$ is equal to the value of $|P(H,-1)|$, where $P(H,lambda)$ is the chromatic polynomial of a hypergraph $H$ which is constructed from $G$. Finally we show that the multiplicity of root $0$ of $P(H,lambda)$ may be at least $2$ for some connected hypergraphs $H$, and the multiplicity of root $1$ of $P(H,lambda)$ may be $1$ for some connected and separable hypergraphs $H$ and may be $2$ for some connected and non-separable hypergraphs $H$.
The binomial Eulerian polynomials, introduced by Postnikov, Reiner, and Williams, are $gamma$-positive polynomials and can be interpreted as $h$-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations. In this paper, we generalize them to $mathbf{s}$-inversion sequences and prove that these new polynomials have only real roots by the method of interlacing polynomials. Three applications of this result are presented. The first one is to prove the real-rootedness of binomial Eulerian polynomials, which confirms a conjecture of Ma, Ma, and Yeh. The second one is to prove that the symmetric decomposition of binomial Eulerian polynomials for colored permutations is real-rooted. Thirdly, our polynomials for certain $mathbf{s}$-inversion sequences are shown to admit a similar geometric interpretation related to edgewise subdivisions of simplexes.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
