Related papers: Existence of Optimally-Greatest Digraphs for Stron…
The present study was concerned with network failure problems for simple connected undirected graphs. A connected graph becomes unconnected through edge failure, under the assumptions that only edges can fail and each edge has an identical…
Given a graph $G$, let $\mathrm{diam}(G)$ be the greatest distance between any two vertices of $G$ which lie in the same connected component, and let $\mathrm{diam}^+(G)$ be the greatest distance between any two vertices of $G$; so…
We improve the best known lower bounds on the exponential behavior of the maximum of the number of connected sets, $N(G)$, and dominating connected sets, $N_{dom}(G)$, for regular graphs. These lower bounds are improved by constructing a…
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…
In this paper, we study a hypothesis test to determine the underlying directed graph structure of nodes in a network, where the nodes represent random processes and the direction of the links indicate a causal relationship between said…
The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of…
The notions of $r$-robustness and $(r,s)$-robustness of a network have been earlier introduced in the literature to achieve resilient consensus in the presence of misbehaving agents. However, while higher robustness levels enable networks…
We study the properties of the giant connected component in random graphs with arbitrary degree distribution. We concentrate on the degree-degree correlations. We show that the adjoining nodes in the giant connected component are correlated…
Let $\Lambda(T)$ denote the set of leaves in a tree $T$. One natural problem is to look for a spanning tree $T$ of a given graph $G$ such that $\Lambda(T)$ is as large as possible. This problem is called maximum leaf number, and it is a…
We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on $n$ vertices. Let $p$ be the edge probability, and write $p=\frac{1+\vep}{2(n-1)}$ for some $\vep\in \R$. In Borgs…
In varying degree distributions, we investigate the optimally robust networks against targeted attacks to nodes with higher degrees. In considering that a network tends to have more robustness with a smaller variance of degree…
Markoff mod-$p$ graphs are conjectured to be connected for all primes $p$. In this paper, we use results of Chen and Bourgain, Gamburd, and Sarnak to confirm the conjecture for all $p > 3.448\cdot10^{392}$. We also provide a method that…
In recent years, there have been intense research efforts to develop efficient methods for probabilistic inference in probabilistic influence diagrams or belief networks. Many people have concluded that the best methods are those based on…
Given a directed acyclic graph (DAG) $G = (V,E)$, we say that $G$ is $(e,d)$-depth-robust (resp. $(e,d)$-edge-depth-robust) if for any set $S \subset V$ (resp. $S \subseteq E$) of at most $|S| \leq e$ nodes (resp. edges) the graph $G-S$…
A temporal graph is a graph in which edges are assigned a time label. Two nodes u and v of a temporal graph are connected one to the other if there exists a path from u to v with increasing edge time labels. We consider the problem of…
Let $G = (V,E)$ be a connected directed graph on $n$ vertices. Assign values from the set $\{1,2,\dots,n\}$ to the vertices of $G$ and update the values according to the following rule: uniformly at random choose a vertex and update its…
An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth…
The modelling of interconnection networks by graphs motivated the study of several extremal problems that involve well known parameters of a graph (degree, diameter, girth and order) and ask for the optimal value of one of them while…
The calculation of network reliability in a probabilistic context has long been an issue of practical and academic importance. Conventional approaches (determination of bounds, sums of disjoint products algorithms, Monte Carlo evaluations,…
Cut vertices are often used as a measure of nodes' importance within a network. They are those nodes whose failure disconnects a graph. Let N(G) be the number of connected induced subgraphs of a graph $G$. In this work, we investigate the…