We consider the discrete Laplace operator ∆ on Erdős–Rényi random graphs with N vertices and edge probability p/N . We are interested in the limiting spectral properties of ∆ as N → ∞ in the subcritical regime 0 < p < 1 where no giant cluster emerges. We prove that in this limit the expectation value of the integrated density of states of ∆ exhibits a Lifshitz-tail behavior at the lower spectra...

We study consistent collections of random fragmentation trees with random integervalued edge lengths. We prove several equivalent necessary and sufficient conditions under which Geometrically distributed edge lengths can be consistently assigned to a Markov branching tree. Among these conditions is a characterization by a unique probability measure, which plays a role similar to the dislocation...

Let G = Gn,n,p be the random bipartite graph on n+n vertices, where each e ∈ [n] appears as an edge independently with probability p. Suppose that each edge e is given an independent uniform exponential rate one cost. Let C(G) denote the expected length of the minimum cost perfect matching. We show that w.h.p. if d = np (log n) then E [C(G)] = (1 + o(1)) 2 6p . This generalises the well-known r...

This report proposes a new image segmentation method with a multi-resolution mechanism. In this method, the high accuracy on image edge extraction and image texture evaluation is realized. The proposed method employs the edge extraction based on the average values of intensities in a small image part and also the texture evaluation based on the co-occurrence probability of a local image area. I...

In this paper, a method for anonymization of social networks by clustering of k-edge-connected subgraphs (SACK) is presented. Previous anonymization algorithms do not consider distribution of nodes in social network graph according to their attributes. SACk tries to focus on this aspect that probability of existence of an edge between two nodes is related to their attributes and this leads to a...

We show that provided log n/n ≤ p ≤ 1 − n−1/4 log n we can with high probability find a collection of bδ(G)/2c edge-disjoint Hamilton cycles in G ∼ Gn,p, plus an additional edge-disjoint matching of size bn/2c if δ(G) is odd. This is clearly optimal and confirms, for the above range of p, a conjecture of Frieze and Krivelevich.

For a positive integer n and an even integer k, let C(n, k) denote the graph with vertex set Figure 1: C(8, 4), newly rewired edges are excluded from future rewire. The Watts-Strogatz graph[2], denoted W S(n, k, p) is obtained from C(n, k) by replacing each edge in C(n, k) with probability p by a randomly chosen edge.

We consider the discrete Laplace operator ∆ on Erdős–Rényi random graphs with N vertices and edge probability p/N . We are interested in the limiting spectral properties of ∆ as N → ∞ in the subcritical regime 0 < p < 1 where no giant cluster emerges. We prove that in this limit the expectation value of the integrated density of states of ∆ exhibits a Lifshits-tail behaviour at the lower spectr...

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graphs and hypergraphs. In particular, we firstly show that for k > 3, if pnk−1/ log n tends to infinity, then a random k-uniform hypergraph on n vertices, with edge probability p, with high probability (w.h.p.) contains a l...

Network reliability is a well-studied problem that requires to measure the probability that a target node is reachable from a source node in a probabilistic (or uncertain) graph, i.e., a graph where every edge is assigned a probability of existence. Many approaches and problem variants have been considered in the literature, all assuming that edge-existence probabilities are fixed. Nevertheless...