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We consider the problem of finding a spanning tree with maximum number of leaves (MaxLeaf). A 2-approximation algorithm is known for this problem, and a 3/2-approximation algorithm when restricted to graphs where every vertex has degree 3…
Networks are ubiquitous in biology and computational approaches have been largely investigated for their inference. In particular, supervised machine learning methods can be used to complete a partially known network by integrating various…
Virtually all network analyses involve structural measures between pairs of vertices, or of the vertices themselves, and the large amount of symmetry present in real-world complex networks is inherited by such measures. This has practical…
The Minimum Branch Vertices Spanning Tree problem aims to find a spanning tree $T$ in a given graph $G$ with the fewest branch vertices, defined as vertices with a degree three or more in $T$. This problem, known to be NP-hard, has…
We study connected graphs with a fixed degree sequence, in the sparse setting where the number of edges grows linearly in the number of vertices. Using the relation to the configuration model, we identify the number of such connected graphs…
We consider a Gibbs distribution over all spanning trees of an undirected, edge weighted finite graph, where, up to normalization, the probability of each tree is given by the product of its edge weights. Defining the weighted degree of a…
Phylogenetic networks generalize phylogenetic trees in order to model reticulation events. Although the comparison of phylogenetic trees is well studied, and there are multiple ways to do it in an efficient way, the situation is much…
Phylogenetic networks are rooted acyclic directed graphs in which the leaves are identified with members of a set X of species. The cluster of a vertex is the set of leaves that are descendants of the vertex. A network is "distinct-cluster"…
In this paper, we give a linear algorithm for obtaining the Laplacian eigenvalues of a cograph. This approach is more efficient as there is no need to directly compute the eigenvalues of Laplacian matrix related to this class of graph. As…
Complex networks are a powerful modeling tool, allowing the study of countless real-world systems. They have been used in very different domains such as computer science, biology, sociology, management, etc. Authors have been trying to…
We prove, that every connected graph with $s$ vertices of degree 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${2\over 5}t +{1\over 5}s+\alpha$ leaves, where $\alpha \ge {8\over 5}$. Moreover, $\alpha \ge 2$ for…
Let $N\geq 2$ be an integer, a (1, $N$)-periodic graph $G$ is a periodic graph whose vertices can be partitioned into two sets $V_1=\{v\mid\sigma(v)=v\}$ and $V_2=\{v\mid\sigma^i(v)\neq v\ \mbox{for any}\ 1<i<N\}$, where $\sigma$ is an…
The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses…
One can often make inferences about a growing network from its current state alone. For example, it is generally possible to determine how a network changed over time or pick among plausible mechanisms explaining its growth. In practice,…
Networks arise in many applications, such as in the analysis of text documents, social interactions and brain activity. We develop a general framework for extrinsic statistical analysis of samples of networks, motivated by networks…
We investigate the properties of the spanning trees of various real-world and model networks. The spanning tree representing the communication kernel of the original network is determined by maximizing total weight of edges, whose weights…
We prove that every connected graph with $s$ vertices of degree~1 and 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${1\over 3}t +{1\over 4}s+{3\over 2}$ leaves. We present infinite series of graphs showing that…
Phylogenetic networks model reticulate evolutionary histories. The last two decades have seen an increased interest in establishing mathematical results and developing computational methods for inferring and analyzing these networks. A…
We explore a simple mathematical model of network computation, based on Markov chains. Similar models apply to a broad range of computational phenomena, arising in networks of computers, as well as in genetic, and neural nets, in social…
In the present paper we consider the problem of constructing all the projective rooted spanning trees of a given graph. We propose an algorithm based on reducing this problem to the problem of constructing all the maximal independent sets…