Related papers: Connectivity threshold for superpositions of Berno…

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…

Given a multigraph $G$, the all-terminal reliability $R(G,p)$ is the probability that $G$ remains connected under percolation with parameter $p$. Fixing the number of vertices $n$ and edges $m$, we investigate which graphs maximize $R(G,p)$…

We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$ on the unit cube…

In this paper we study the connectivity properties of the random subgraph of the $n$-cube generated by the $k$-out model and denoted by $Q^n(k)$. Let $k$ be an integer, $1\leq k \leq n-1$. We let $Q^n(k)$ be the graph that is generated by…

Let $\mathcal{P}$ be a set of $n=2m+1$ points in the plane in general position. We define the graph $GM_\mathcal{P}$ whose vertex set is the set of all plane matchings on $\mathcal{P}$ with exactly $m$ edges. Two vertices in…

A path in an edge-colored graph is called a \emph{rainbow path} if all edges on it have pairwise distinct colors. For $k\geq 1$, the \emph{rainbow-$k$-connectivity} of a graph $G$, denoted $rc_k(G)$, is the minimum number of colors required…

We present new results concerning threshold functions for a wide family of random intersection graphs. To this end we apply the coupling method used for establishing threshold functions for homogeneous random intersection graphs introduced…

In interconnection networks, matching preclusion is a measure of robustness when there is a link failure. Let $G$ be a graph of even order. The matching preclusion number $mp(G)$ is defined as the minimum number of edges whose deletion…

Let $\{G_n\}_{n=1}^{\infty}$ be a sequence of transitive infinite connected graphs with $\sup\limits_{n\geq 1} p_c(G_n) < 1,$ where each $p_c(G_n)$ is bond percolation critical probability on $G_n.$ Schramm (2008) conjectured that if $G_n$…

We prove that the heavy clusters are indistinguishable for Bernoulli percolation on quasi-transitive nonunimodular graphs. As an application, we show that the uniqueness threshold of any quasi-transitive graph is also the threshold for…

For a graph $G$ and for two distinct vertices $u$ and $v$, let $\kappa(u,v)$ be the maximum number of vertex-disjoint paths joining $u$ and $v$ in $G$. The average connectivity matrix of an $n$-vertex connected graph $G$, written…

A layered graph $G^\times$ is the Cartesian product of a graph $G = (V,E)$ with the linear graph $Z$, e.g. $Z^\times$ is the 2D square lattice $Z^2$. For Bernoulli percolation with parameter $p \in [0,1]$ on $G^\times$ one intuitively would…

In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on $[n]=\{1,2,\ldots,n\}$ with $m$ edges, whenever $n\to\infty$ and $n-1\le m=m(n)\le \binom{n}{2}$. We give an asymptotic formula for the…

For each finite simple graph $G$, Aharoni, Berger and Ziv consider a recursively defined number $\psi (G) \in \mathbb{Z}\cup \{+ \infty \}$ which gives a lower bound for the topological connectivity of the independence complex $I_G$. They…

k-connectivity of random graphs is a fundamental property indicating reliability of multi-hop wireless sensor networks (WSN). WSNs comprising of sensor nodes with limited power resources are modeled by random graphs with unreliable nodes,…

Let $G=G_{n,k}$ denote the graph formed by placing points in a square of area $n$ according to a Poisson process of density 1 and joining each pair of points which are both $k$ nearest neighbours of each other. Then $G_{n,k}$ can be used as…

For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$…

In this paper, we have studied various mixed distributions generated by two uniform distributions: first, where the supports are two connected line segments, and second, where the supports are two disconnected line segments. For these mixed…

This paper provides a necessary and sufficient condition for a random network with nodes Poissonly distributed on a unit square and a pair of nodes directly connected following a generic random connection model to be asymptotically almost…

Let $q(G)$ denote the $Q$-index of a graph $G$, which is the largest signless Laplacian eigenvalue of $G$. We prove best possible upper bounds of $q(G)$ and best possible lower bounds of $q(\overline{G})$ for a connected graph $G$ to be…