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We study a fading linear finite-field relay network having multiple source-destination pairs. Because of the interference created by different unicast sessions, the problem of finding its capacity region is in general difficult. We observe…
The rich and varied ways that genetic material can be passed between species has motivated extensive research into the theory of phylogenetic networks. Features that align with biological processes, or with desirable mathematical…
We introduce and investigate the solvable graph $\Gamma_\mathfrak{S}(L)$ of a finite-dimensional Lie algebra $L$ over a field $F$. The vertices are the elements outside the solvabilizer $\sol(L)$, and two vertices are adjacent whenever they…
It is known that there exists a network, called as the M-network, which is not scalar linearly solvable but has a vector linear solution for message dimension two. Recently, a generalization of this result has been presented where it has…
Normal networks are an important class of phylogenetic networks that have compelling mathematical properties which align with intuition about inference from genetic data. While tools enabling widespread use of phylogenetic networks in the…
A large class of phylogenetic networks can be obtained from trees by the addition of horizontal edges between the tree edges. These networks are called tree based networks. Reticulation-visible networks and child-sibling networks are all…
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…
We study the problem of communicating over a single-source single-terminal network in the presence of an adversary that may jam a single link of the network. If any one of the edges can be jammed, the capacity of such networks is well…
We consider the multiple unicast problem under network coding over directed acyclic networks with unit capacity edges. There is a set of n source-terminal (s_i - t_i) pairs that wish to communicate at unit rate over this network. The…
While feasibility and obtaining a solution of a given network coding problem are well studied, the decoding procedure and complexity have not garnered much attention. We consider the decoding problem in a network wherein the sources…
Parties connected to independent sources through a network can generate correlations among themselves. Notably, the space of feasible correlations for a given network, depends on the physical nature of the sources and the measurements…
Phylogenetic networks are a generalization of phylogenetic trees that are used to represent non-tree-like evolutionary histories that arise in organisms such as plants and bacteria, or uncertainty in evolutionary histories. An…
In a graph, we assign distinct integers to the vertices, and take the sum of two integers if they are on two adjacent vertices. The minimum possible number of different sums is the \emph{sum index} of this graph. In this paper, we present…
Networks constitute efficient tools for assessing universal features of complex systems. In physical contexts, classical as well as quantum, networks are used to describe a wide range of phenomena, such as phase transitions, intricate…
Graphs are called navigable if one can find short paths through them using only local knowledge. It has been shown that for a graph to be navigable, its construction needs to meet strict criteria. Since such graphs nevertheless seem to…
Le n be any positive integer. A hyperbinary expansion of n is are presentation of n as sum of powers of 2, each power being used at most twice. In this paper we study some properties of a suitable edge-coloured and vertex-weighted oriented…
We develop a general finite-alphabet framework for Euler-type sums based on the notion of a monoidal alphabet. An alphabet of summand letters is called monoidal when it is closed under pointwise multiplication, thereby inducing the usual…
Let $G$ be a finite group and construct a graph $\Delta(G)$ by taking $G\setminus\{1\}$ as the vertex set of $\Delta(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set…
Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…
We study a certain polytope depending on a graph $G$ and a parameter $\beta\in(0,1)$ which arises from embedding the Hamiltonian cycle problem in a discounted Markov decision process. Eshragh \emph{et al.} conjectured a lower bound on the…