Related papers: On the Growth of a Ballistic Deposition Model on F…

In this work we establish under certain hypotheses the $N \to +\infty$ asymptotic expansion of integrals of the form $$\mathcal{Z}_{N,\Gamma}[V] \, = \, \int_{\Gamma^N} \prod_{ a < b}^{N}(z_a - z_b)^\beta \, \prod_{k=1}^{N} \mathrm{e}^{ - N…

The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhood covers all the vertices of the graph. The packing number $\rho(G)$ of $G$ is the maximum size of a vertex set whose closed…

We study the asymptotics of large, moderate and normal deviations for the connected components of the sparse random graph by the method of stochastic processes. We obtain the logarithmic asymptotics of large deviations of the joint…

We consider the adjacency matrix $A$ of a large random graph and study fluctuations of the function $f_n(z,u)=\frac{1}{n}\sum_{k=1}^n\exp\{-uG_{kk}(z)\}$ with $G(z)=(z-iA)^{-1}$. We prove that the moments of fluctuations normalized by…

The investigation asymptotic limits on associated data mainly focused on limit theorems of summands of associated data and on the related invariance principles. In a series of papers, we are going to set the general frame of the theory by…

Let $H$ be a graph on $n$ vertices and let the blow-up graph $G[H]$ be defined as follows. We replace each vertex $v_i$ of $H$ by a cluster $A_i$ and connect some pairs of vertices of $A_i$ and $A_j$ if $(v_i,v_j)$ was an edge of the graph…

In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma\_{p,k}(G-e)$,…

In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. Time dependence of metric represents the evolution of the stress-free (natural)…

In this paper, we establish the asymptotic expansion at infinity of gradient graph in dimension 2 with vanishing mean curvature at infinity. This corresponds to our previous results in higher dimensions and generalizes the results for…

We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of…

In this paper we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott, and Goldman. Let $\Sigma_{g}$ denote a…

Let $k\ge 1$ be an odd integer, $t=\lfloor {{k+2}\over 4}\rfloor$, and $q$ be a prime power. We construct a bipartite, $q$-regular, edge-transitive graph $C\!D(k,q)$ of order $v \le 2q^{k-t+1}$ and girth $g \ge k+5$. If $e$ is the the…

In the vertex connectivity augmentation problem, we are given an undirected $n$-vertex graph $G$, a set of links $L \subseteq \binom{V(G)}{2} \setminus E(G)$, and integers $\lambda$ and $k$. The task is to insert at most $k$ links from $L$…

A random graph evolution based on the interactions of N vertices is studied. During the evolution both the preferential attachment method and the uniform choice of vertices are allowed. The weight of a vertex means the number of its…

We prove a central limit theorem for a certain class of functions on sparse rank-one inhomogeneous random graphs endowed with additional i.i.d. edge and vertex weights. Our proof of the central limit theorem uses a perturbative form of…

The interest is in models of integer-valued height functions on shift-invariant planar graphs whose maximum degree is three. We prove delocalisation for models induced by convex nearest-neighbour potentials, under the condition that each…

We introduce graph width parameters, called $\alpha$-edge-crossing width and edge-crossing width. These are defined in terms of the number of edges crossing a bag of a tree-cut decomposition. They are motivated by edge-cut width, recently…

A \emph{$t$-treewidth-modulator} of a graph $G$ is a set $X \subseteq V(G)$ such that the treewidth of $G-X$ is at most some constant $t-1$. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs $G$ that…

We study the free product of rooted graphs and its various decompositions using quantum probabilistic methods. We show that the free product of rooted graphs is canonically associated with free independence, which completes the proof of the…

We study the evolution of the susceptibility in the subcritical random graph $G(n,p)$ as $n$ tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the…