We consider undirected graphs, and assume that each edge of G exists with probability p 2 (0; 1). The all{terminal reliability function A(G; p) of such a random graph G is the probability that the spanning subgraph formed by the existing edges is connected. A two{graph is a graph with two distinguished vertices called terminals. The two{terminal reliability function T (G; p) of a two{graph G is...

In papers[1, 2], the amplitudes of continuous-time quantum walk on graphs possessing quantum decomposition (QD graphs) have been calculated by a new method based on spectral distribution associated to their adjacency matrix. Here in this paper, it is shown that the continuous-time quantum walk on any arbitrary graph can be investigated by spectral distribution method, simply by using Krylov sub...

We study how the outcome of evolutionary dynamics on graphs depends on a randomness on the graph structure. We gradually change the underlying graph from completely regular (e.g. a square lattice) to completely random. We find that the fixation probability increases as the randomness increases; nevertheless, the increase is not significant and thus the fixation probability could be estimated by...

We show that for random bit strings, Up(n), with probability, p = 12 , the firstorder quantifier depth D(Up(n)) needed to distinguish non-isomorphic structures is Θ(lg lg n), with high probability. Further, we show that, with high probability, for random ordered graphs, G≤,p(n) with edge probabiltiy p = 12 , D(G≤,p(n)) = Θ(log∗ n), contrasting with the results of random (non-ordered) graphs, Gp...

Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d− 1)2g−1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresp...

One of the generalizations of the Wiener number to weighted graphs is to assign probabilities to edges, meaning that in nonstatic conditions the edge is present only with some probability. The Reliability Wiener number is defined as the sum of reliabilities among pairs of vertices, where the reliability of a pair is the reliability of the most reliable path. Closed expressions are derived for t...

An accurate approximate formula of the die-out probability in a SIS epidemic process on a network is proposed. The formula contains only three essential parameters: the largest eigenvalue of the adjacency matrix of the network, the effective infection rate of the virus, and the initial number of infected nodes in the network. The die-out probability formula is compared with the exact die-out pr...

Formally, when we are given a graph G and we say this is a random graph, we are wrong. A given graph is fixed, there is nothing random to it. What we mean though through this term abuse is that this graph was sampled out of a set of graphs according to a probability distribution. For instance, Figure 1.1 shows the three possible graphs on vertex set [3] = {1, 2, 3} with 2 edges. The probability...

We consider events over the probability space generated by the degree sequences of two independent Erdős-Rényi random graphs, and consider an approximation probability space where such degree sequences are deemed to be sequences of i.i.d. random variables. We show that, for any sequence of events with probabilities asymptotically smaller than some power law in the approximation model, the same ...