Related papers: The connectivity-profile of random increasing k-tr…
Betweenness centrality of a vertex in a graph measures the fraction of shortest paths going through the vertex. This is a basic notion for determining the importance of a vertex in a network. The k-betweenness centrality of a vertex is…
We propose a network characterization of combinatorial fitness landscapes by adapting the notion of inherent networks proposed for energy surfaces. We use the well-known family of NK landscapes as an example. In our case the inherent…
The use of cumulative incidence functions for characterizing the risk of one type of event in the presence of others has become increasingly popular over the past decade. The problems of modeling, estimation and inference have been treated…
The giant $k$-core --- maximal connected subgraph of a network where each node has at least $k$ neighbors --- is important in the study of phase transitions and in applications of network theory. Unlike Erd\H{o}s-R\'enyi graphs and other…
Phylogenetic networks generalize phylogenetic trees, and have been introduced in order to describe evolution in the case of transfer of genetic material between coexisting species. There are many classes of phylogenetic networks, which can…
Designing well-connected graphs is a fundamental problem that frequently arises in various contexts across science and engineering. The weighted number of spanning trees, as a connectivity measure, emerges in numerous problems and plays a…
We propose a network characterization of combinatorial fitness landscapes by adapting the notion of inherent networks proposed for energy surfaces. We use the well-known family of NK landscapes as an example. In our case the inherent…
A vertex of degree one is called an end-vertex, and an end-vertex of a tree is called a leaf. A tree with at most $k$ leaves is called a $k$-ended tree. For a positive integer $k$, let $t_k$ be the order of a largest $k$-ended tree. Let…
We propose a general approach to the description of spectra of complex networks. For the spectra of networks with uncorrelated vertices (and a local tree-like structure), exact equations are derived. These equations are generalized to the…
In this paper we establish all extremal graphs with respect to augmented eccentric connectivity index among all (simple connected) graphs, among trees and among trees with perfect matching. For graphs that turn out to be extremal explicit…
We consider a model of random tree growth, where at each time unit a new vertex is added and attached to an already existing vertex chosen at random. The probability with which a vertex with degree $k$ is chosen is proportional to $w(k)$,…
We establish limit theorems that describe the asymptotic local and global geometric behaviour of random enriched trees considered up to symmetry. We apply these general results to random unlabelled weighted rooted graphs and uniform random…
Statistical properties of evolving random graphs are analyzed using kinetic theory. Treating the linking process dynamically, structural characteristics such as links, paths, cycles, and components are obtained analytically using the rate…
We study the class of graphs known as k-trees through the lens of Joyal's theory of combinatorial species (and an equivariant extension known as '$\Gamma$-species' which incorporates data about 'structural' group actions). This culminates…
Two previous papers, arXiv:1803.00284 and arXiv:1803.00281, introduced and studied strong subgraph $k$-connectivity of digraphs obtaining characterizations, lower and upper bounds and computational complexity results for the new digraph…
Topology of exponential and scale-free trees and simple graphs is investigated numerically. The numbers of the nearest neighbors, the next-nearest neighbors, the next-next-nearest neighbors, the 4-th and the 5-th neighbors are calculated.…
It has been recently proposed that the natural connectivity can be used to characterize efficiently the robustness of complex networks. The natural connectivity quantifies the redundancy of alternative routes in the network by evaluating…
The resilience of a complex interconnected system concerns the size of the macroscopic functioning node clusters after external perturbations based on a random or designed scheme. For a representation of the interconnected systems with…
For a connected graph, a {\em minimum vertex separator} is a minimum set of vertices whose removal creates at least two connected components. The vertex connectivity of the graph refers to the size of the minimum vertex separator and a…
The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$ is a parameter that can measure the reliability of a network $G$ to connect any $k$ vertices in $G$, which is proved to be NP-complete for a general graph $G$. Let $S\subseteq…