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is dual to a red edge inside the region R . Since T is a tree , we can enumerate its vertices v1,v2,…,vn so that v1 is a leaf of the tree (vertex of degree one), and the subgraph induced by any {v1,…,vk ...
$$ vertices yields equation ( 1 ). 1.3 Treewidth and maximum degree Bagrow makes a study of the modularity of some trees and treelike graphs in [6]. He shows that Galton–Watson trees and $$ k $$-ary trees have ...
that for all |$ 1 \leqslant j ...
|$t\in\mathbb{R}$| there exists |$s\le t$| such that the inclusion |$Y^{t\le h_\chi} \to Y^{s\le h_\chi}$| induces the trivial map in |$\pi_k$| for all |$ k \le m- 1 $|.) □ Here |$Y^{t\le h_\chi ...
topological properties of the nodes for each sub-network: The average node degree |$\langle k \rangle_j = \sum_{j \in \mathcal{G}_i} k_j /|G_i|$| for |$j$|= 1 ..,|$G,$| where |$|G_i|$| is the number of nodes ...
t$| if and only if there exists a forest |$f$| such that |$f\circ s=t$|. Consider the set of pairs of trees |$(t,s)$|, where |$t$| and |$s$| have the same number of leaves, that we mod out ...
to |${\textrm{Id}}$| modulo |$n$|, and choosing |$e$| large , we may assume that the entries of |$f$| satisfy |$e> c > d > a > b > 1 $|. Then, a simple recursion shows that the degree of |$(f\circ h)^ k $| is equal ...
topology) |$\operatorname{Map}(\Delta ^p, X^p)$| for all |$p\geq 0$| containing those maps that commute with the cofaces and the codegeneracies. We have an evaluation map $$\begin{align}\operatorname{ev}_p ...
vertices |$u$| and |$v$|, if |$|N(u)\cap N(v)|>0$|, then |$|N(u)\cap N(v)|=2$|. (2) Let |$S$| be a subgraph of |$Q_n$| with |$|S|= k $| and |$ k \leq n+ 1 $|. Then |$|N_{Q_n}(S)|\geq kn-\frac{ k ( k + 1 )}{2}+ 1 ...
}}$$ and $$(U_{ k })_{ k \in \mathbb Z_{\geq 0}}$$ simply as Chebyshev polynomials, see also Remark 2.12 below. Proposition 2.10. [22, § 2.3] For all $$ k \geq 1 ,$$ the following identities hold: \begin{align*} x^ k &=T_k ...
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