Random graphs are graphs with a fixed number of vertices, where each edge is present with a fixed probability. We are interested in the probability that a random graph contains a certain pattern, for example a cycle or a clique. A very high edge probability gives rise to perhaps too many edges (which degrades performance for many algorithms), whereas a low edge probability might result in a dis...

Abstract We use tools from free probability to study the spectra of Hermitian operators on infinite graphs. Special attention is devoted universal covering trees finite For these graphs, we derive a new variational formula for spectral radius and provide proofs results due Sunada Aomoto using probability. With goal extending applicability techniques beyond trees, introduce combinatorial product...

let $n$ be any positive integer and let $f_n$ be the friendship (or dutch windmill) graph with $2n+1$ vertices and $3n$ edges. here we study graphs with the same adjacency spectrum as the $f_n$. two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. let $g$ be a graph cospectral with $f_n$. here we prove that if $g$ has no cycle of length $4$ or $...

A classical theorem by Erdős, Kleitman and Rothschild on the structure of triangle-free graphs states that with high probability such graphs are bipartite. Our first main result refines this theorem by investigating the structure of the ’few’ triangle-free graphs which are not bipartite. We prove that with high probability these graphs are bipartite up to a few vertices. Similar results hold if...

We say that a graph G = (V,E) on n vertices is a β-expander for some constant β > 0 if every U ⊆ V of cardinality |U | ≤ n 2 satisfies |NG(U)| ≥ β|U | where NG(U) denotes the neighborhood of U . We explore the process of uniformly at random deleting vertices of a β-expander with probability n for some constant α > 0. Our main result implies that as n tends to infinity, the deletion process perf...

We consider the modified Moran process on graphs to study the spread of genetic and cultural mutations on structured populations. An initial mutant arises either spontaneously (aka uniform initialization), or during reproduction (aka temperature initialization) in a population of n individuals, and has a fixed fitness advantage r > 1 over the residents of the population. The fixation probabilit...

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, which is known as the node fault model. In this paper, we investigate k-connectivity of random graphs in the node fault model by evaluating the network breakd...

We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose. We also show that in the k-nearest neighbour model, there is a constant κ such that almost every κ-connected graph has a Hamilton cycle.

Let Ω be a sample space. We are going to assume that Ω is either a finite set {ω1, . . . , ωn}, or infinite countable set {ωi, i ∈ N}, where N = {1, . . . , } is the set of positive integers. The two kinds of such sets are called countable. Denote by #Ω the cardinality of Ω. Thus #Ω = n ∈ N if Ω is a finite set. #Ω = א0 if Ω is infinite countable. To each ω ∈ Ω we attach a probability (mass) p(...