Related papers: Minimality of Tree Tensor Network Ranks
What do generic networks that have certain properties look like? We define Relative Canonical Network ensembles as the ensembles that realize a property R while being as indistinguishable as possible from a generic network ensemble. This…
A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives…
We employ matrix product states (MPS) and tensor networks to study topological properties of the space of ground states of gapped many-body systems. We focus on families of states in one spatial dimension, where each state can be…
A $d$-dimensional tensegrity framework $(T,p)$ is an edge-labeled geometric graph in ${\mathbb R}^d$, which consists of a graph $T=(V,B\cup C\cup S)$ and a map $p:V\to {\mathbb R}^d$. The labels determine whether an edge $uv$ of $T$…
For $S\subseteq V(G)$ and $|S|\geq 2$, $\lambda(S)$ is the maximum number of edge-disjoint trees connecting $S$ in $G$. For an integer $k$ with $2\leq k\leq n$, the \emph{generalized $k$-edge-connectivity} $\lambda_k(G)$ of $G$ is then…
Recently there has been considerable interest in the problem of finding a phylogenetic network with a minimum number of reticulation vertices which displays a given set of phylogenetic trees, that is, a network with minimum hybrid number.…
We answer to a question posed recently by B. Lovitz and F. Petrov, proving the conjectured sufficient minimality and uniqueness condition of the 3-tensor decomposition.
A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. This paper mostly deals with the topological group $H_+(X)$ of order-preserving homeomorphisms of a compact linearly ordered…
For a given set $\mathcal{L}$ of species and a set $\mathcal{T}$ of triplets on $\mathcal{L}$, one wants to construct a phylogenetic network which is consistent with $\mathcal{T}$, i.e which represents all triplets of $\mathcal{T}$. The…
Random tensor networks are a powerful toy model for understanding the entanglement structure of holographic quantum gravity. However, unlike holographic quantum gravity, their entanglement spectra are flat. It has therefore been argued that…
In this article the notion of the nondecreasing (ND) rank of a matrix or tensor is introduced. A tensor has an ND rank of r if it can be represented as a sum of r outer products of vectors, with each vector satisfying a monotonicity…
Modularity is a very widely used measure of the level of clustering or community structure in networks. Here we consider a recent generalisation of the definition of modularity to temporal graphs, whose edge-sets change over discrete…
The atom-bond connectivity (ABC) index is one of the most investigated degree-based molecular structure descriptors with a variety of chemical applications. It is known that among all connected graphs, the trees minimize the ABC index.…
In many cases, Neural networks can be mapped into tensor networks with an exponentially large bond dimension. Here, we compare different sub-classes of neural network states, with their mapped tensor network counterpart for studying the…
We introduce the graph theoretical parameter of edge treewidth. This parameter occurs in a natural way as the tree-like analogue of cutwidth or, alternatively, as an edge-analogue of treewidth. We study the combinatorial properties of…
We study the minimal spanning arborescence which is the directed analogue of the minimal spanning tree, with a particular focus on its infinite volume limit and its geometric properties. We prove that in a certain large class of transient…
We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a…
Let $G$ be a nontrivial connected graph of order $n$ and $k$ an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such…
We introduce the concept of Most, and Least, Compact Spanning Trees - denoted respectively by $T^*(G)$ and $T^\#(G)$ - of a simple, connected, undirected and unweighted graph $G(V, E, W)$. For a spanning tree $T(G) \in \mathcal{T}(G)$ to be…
The hierarchical (multi-linear) rank of an order-$d$ tensor is key in determining the cost of representing a tensor as a (tree) Tensor Network (TN). In general, it is known that, for a fixed accuracy, a tensor with random entries cannot be…