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      Questions tagged [trees]

      A tree is a connected graph without cycles, with a finite or infinite number of vertices. There are many variants of trees, according to further constraints or decorations.

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      Groups acting on trees

      Assume that $X$ is a tree such that every vertex has infinite degree, and a discrete group $G$ acts on this tree properly (with finite stabilizers) and transitively. Is it true that $G$ contains a ...
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      Treewidth problem equivalence

      Say we are solving a tree decomposition problem, e.g. given a graph $G = (V, E)$ we try to find a chordal graph $H$ such that $V(H) = V(G)$, $E(G) \in E(H)$ and the maximal clique in $H$ is minimal ...
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      30 views

      Admissible (unitary) spherical representation $sl(2,Q_p)$. Does dimension fixed point vector increase proportional index

      I am using terminology of Cartier's Harmonic analysis on trees. Take $\pi$ be one of the irreducible principal or complementary (unitary) spehrical series of $Sl(2, Q_p)$. Let $K$ be the maximal ...
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      votes
      2answers
      592 views

      Terminology about trees

      In set theory, a tree is usually defined as a partial order such that the set of elements below any given one is well-ordered. I am interested in the class of partial orders $P$ such that for every $...
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      67 views

      Distributions of “sequential” binomials

      I have come across the following stochastic process which seems very elementary, although I do not know any appropriate terminology for it; I greatly appreciate any suggestions! Suppose I am given ...
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      votes
      1answer
      417 views

      Destroying Suslin, nothing special

      Recall that a tree on $\omega_1$ is called Suslin if every chain and antichain are countable. If every level is countable and there are no cofinal branches, then it is called Aronszajn (in particular ...
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      1answer
      74 views

      Typical labelled vs. unlabelled trees properties

      Consider two random tree models $T_1(n)$ and $T_2(n)$, chosen equiprobably among labelled and unlabelled trees on $n$ vertices respectively. I'm wondering if there are properties that are vastly more ...
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      2answers
      298 views

      Almost graceful tree conjecture

      The graceful tree conjecture is the following statement: for any tree $T = (V, E)$ with $|V| = n$ there is a bijective map $f: V \to [n]$ such that $D = \{|f(x) - f(y)| \mid xy \in E\} = [n - 1]$. ...
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      1answer
      251 views

      A monad that unions sets

      Suppose we have a monad that maps types of some kind to other types (see below) , and let types be sets. Let $\alpha, \beta$ be types, $\rightarrow$ denote a function between types, and let $a : \...
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      votes
      1answer
      120 views

      Are there Prüfer sequences for rooted forests?

      One well-known, extremely slick proof of Cayley's tree enumeration theorem is the use of Prüfer sequences. Cayley also proved a version for forests, namely that the number of forests with $n$ ...
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      votes
      1answer
      110 views

      Two disjoint trees

      Let $G$ be a graph and let $A_1, A_2 \subseteq V(G)$ be disjoint sets of vertices. Let us call $(A_1, A_2)$ independent if there exist vertex-disjoint trees $T_1, T_2 \subseteq G$ within $G$ which ...
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      74 views

      Partitioning the vertices of a graph into induced trees

      I am looking for previous work regarding graphs whose vertices can be assigned colours (not necessarily a proper colouring) in such a way that each colour class induces a tree. In particular I am ...
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      0answers
      33 views

      The number of Laplacian eigenvalues of a graph in interval [k,n]

      There are several upper and lower bounds for $m_G[2,n]$ (the number of Laplacian eigenvalues of a graph $G$ with $n$ vertices in the interval $[2,n]$). I want to know whether there exists any bound ...
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      votes
      1answer
      182 views

      Counting promenades on graphs

      Let a "promenade" on a tree be a walk going through every edge of the tree at least once, and such that the starting point and endpoint of the walk are distinct. What we mean by isomorphic promenades ...
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      42 views

      Shattering/covering finite trees, and a simple looking inequality

      Consider the tree $T$ with the set of its maximal elements (denoted $[T]$) equal to $\prod_{m\leq n} X_m$ for some finite sets $X_0,..., X_n$. Let $p(T)=\{b: b\in \prod_{i\leq m \leq j } X_{m}\text{ ...

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