For a finite group
G
, denote by
α
(
G
)
the minimum number of vertices of any graph
Γ
having
Aut
(
Γ
)
≅
G
. In this paper, we prove that
α
(
G
)
≤
|
G
|
, with specified exceptions. The exceptions ...include four infinite families of groups, and 17 other small groups. Additionally, we compute
α
(
G
)
for the groups
G
such that
α
(
G
)
>
|
G
|
where the value
α
(
G
)
was previously unknown.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
The Ehrhart polynomial ehr(P)(n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f*-vector of P, introduced by Felix Breuer in 2012, is the vector of ...coefficients of ehr(P)(n) with respect to the binomial coefficient basis {((n-1)(0)),((n-1)(1)),& mldr;,((n-1)(d))}, where d = dim P. Similarly to h/h*-vectors, the f*-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f*-vectors of lattice polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f*-coefficients increases and the last quarter decreases. Even though f*-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h*-vector, there is a polytope with the same h*-vector whose f*-vector is unimodal.
The Ehrhart polynomial ehr
) of a lattice polytope
counts the number of integer points in the
-th dilate of
. The
-vector of
, introduced by Felix Breuer in 2012, is the vector of coefficients of ehr
...) with respect to the binomial coefficient basis
where
= dim
. Similarly to
-vectors, the
-vector of
coincides with the
-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of
-vectors of lattice polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of
-vectors of simplicial polytopes; e.g., the first half of the
-coefficients increases and the last quarter decreases. Even though
-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart
-vector, there is a polytope with the same
-vector whose
-vector is unimodal.
The Ehrhart polynomial ehrP(n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f∗-vector of P, introduced by Felix Breuer in 2012, is the vector of ...coefficients of ehrP(n) with respect to the binomial coefficient basis (Formula presented.), where d = dimP. Similarly to h/h∗-vectors, the f∗-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f∗-vectors of polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f∗-coefficients increases and the last quarter decreases. Even though f∗-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h∗-vector, there is a polytope with the same h∗-vector whose f∗-vector is unimodal.
For a finite group \(G\), denote by \(\alpha(G)\) the minimum number of vertices of any graph \(\Gamma\) having \(\text{Aut}(\Gamma)\cong G\). In this paper, we prove that \(\alpha(G)\leq |G|\), with ...specified exceptions. The exceptions include four infinite families of groups, and 17 other small groups. Additionally, we compute \(\alpha(G)\) for the groups \(G\) such that \(\alpha(G)> |G|\) where the value \(\alpha(G)\) was previously unknown.
We consider the problem of estimating the marginal independence structure of a Bayesian network from observational data, learning an undirected graph we call the unconditional dependence graph. We ...show that unconditional dependence graphs of Bayesian networks correspond to the graphs having equal independence and intersection numbers. Using this observation, a Gr\"obner basis for a toric ideal associated to unconditional dependence graphs of Bayesian networks is given and then extended by additional binomial relations to connect the space of all such graphs. An MCMC method, called GrUES (Gr\"obner-based Unconditional Equivalence Search), is implemented based on the resulting moves and applied to synthetic Gaussian data. GrUES recovers the true marginal independence structure via a penalized maximum likelihood or MAP estimate at a higher rate than simple independence tests while also yielding an estimate of the posterior, for which the \(20\%\) HPD credible sets include the true structure at a high rate for data-generating graphs with density at least \(0.5\).
Panhandle matroids are a specific lattice-path matroid corresponding to panhandle-shaped Ferrers diagrams. Their matroid polytopes are the subpolytopes carved from a hypersimplex to form matroid ...polytopes of paving matroids. It has been an active area of research to determine which families of matroid polytopes are Ehrhart positive. We prove Ehrhart positivity for panhandle matroid polytopes, thus confirming a conjecture of Hanely, Martin, McGinnis, Miyata, Nasr, Vindas-Meléndez, and Yin (2023). Another standing conjecture posed by Ferroni (2022) asserts that the coefficients of the Ehrhart polynomial of a connected matroid are bounded above by those of the corresponding uniform matroid. We prove Ferroni's conjecture for paving matroids -- a class conjectured to asymptotically contain all matroids. Our proofs rely solely on combinatorial techniques which involve determining intricate interpretations of certain set partitions.
The Ehrhart polynomial $\text{ehr}_P(n)$ of a lattice polytope $P$ counts the
number of integer points in the $n$-th integral dilate of $P$. The $f^*$-vector
of $P$, introduced by Felix Breuer in ...2012, is the vector of coefficients of
$\text{ehr}_P(n)$ with respect to the binomial coefficient basis $
\left\{\binom{n-1}{0},\binom{n-1}{1},...,\binom{n-1}{d}\right\}$, where $d =
\dim P$. Similarly to $h/h^*$-vectors, the $f^*$-vector of $P$ coincides with
the $f$-vector of its unimodular triangulations (if they exist). We present
several inequalities that hold among the coefficients of $f^*$-vectors of
polytopes. These inequalities resemble striking similarities with existing
inequalities for the coefficients of $f$-vectors of simplicial polytopes; e.g.,
the first half of the $f^*$-coefficients increases and the last quarter
decreases. Even though $f^*$-vectors of polytopes are not always unimodal,
there are several families of polytopes that carry the unimodality property. We
also show that for any polytope with a given Ehrhart $h^*$-vector, there is a
polytope with the same $h^*$-vector whose $f^*$-vector is unimodal.
We consider the problem of characterizing Bayesian networks up to unconditional equivalence, i.e., when directed acyclic graphs (DAGs) have the same set of unconditional \(d\)-separation statements. ...Each unconditional equivalence class (UEC) is uniquely represented with an undirected graph whose clique structure encodes the members of the class. Via this structure, we provide a transformational characterization of unconditional equivalence; i.e., we show that two DAGs are in the same UEC if and only if one can be transformed into the other via a finite sequence of specified moves. We also extend this characterization to the essential graphs representing the Markov equivalence classes (MECs) in the UEC. UECs partition the space of MECs and are easily estimable from marginal independence tests. Thus, a characterization of unconditional equivalence has applications in methods that involve searching the space of MECs of Bayesian networks.
