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The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph changes dramatically at $lambda_n=1/binom{n+1}{2}$. For $lambda_n=(1-epsilon)/binom{n+1}{2}$, the random graph consists of components of size at most $O(nln(n))$ a.s. and for $(1+epsilon)/binom{n+1}{2}$, there emerges a unique largest component of size $sim wp(epsilon) cdot 2^ncdot n$!$ a.s.. This giant component is furthermore dense in the reversal graph.
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Given a hereditary property of graphs $mathcal{H}$ and a $pin [0,1]$, the edit distance function ${rm ed}_{mathcal{H}}(p)$ is asymptotically the maximum proportion of edge-additions plus edge-deletions applied to a graph of edge density $p$ sufficient to ensure that the resulting graph satisfies $mathcal{H}$. The edit distance function is directly related to other well-studied quantities such as the speed function for $mathcal{H}$ and the $mathcal{H}$-chromatic number of a random graph. Let $mathcal{H}$ be the property of forbidding an ErdH{o}s-R{e}nyi random graph $Fsim mathbb{G}(n_0,p_0)$, and let $varphi$ represent the golden ratio. In this paper, we show that if $p_0in [1-1/varphi,1/varphi]$, then a.a.s. as $n_0toinfty$, begin{align*} {rm ed}_{mathcal{H}}(p) = (1+o(1)),frac{2log n_0}{n_0} cdotminleft{ frac{p}{-log(1-p_0)}, frac{1-p}{-log p_0} right}. end{align*} Moreover, this holds for $pin [1/3,2/3]$ for any $p_0in (0,1)$.
We consider the generalized game Lights Out played on a graph and investigate the following question: for a given positive integer $n$, what is the probability that a graph chosen uniformly at random from the set of graphs with $n$ vertices yields a universally solvable game of Lights Out? When $n leq 11$, we compute this probability exactly by determining if the game is universally solvable for each graph with $n$ vertices. We approximate this probability for each positive integer $n$ with $n leq 87$ by applying a Monte Carlo method using 1,000,000 trials. We also perform the analogous computations for connected graphs.
We study the distributional properties of horizontal visibility graphs associated with random restrictive growth sequences and random set partitions of size $n.$ Our main results are formulas expressing the expected degree of graph nodes in terms of simple explicit functions of a finite collection of Stirling and Bernoulli numbers.
In this article we consider the cycle structure of compositions of pairs of involutions in the symmetric group S_n chosen uniformly at random. These can be modeled as modified 2-regular graphs, giving rise to exponential generating functions. A composition of two random involutions in S_n typically has about n^(1/2) cycles, and the cycles are characteristically of length n^(1/2). Compositions of two random fixed-point-free involutions, on the other hand, typically have about log n cycles and are closely related to permutations with all cycle lengths even. The number of factorizations of a random permutation into two involutions appears to be asymptotically lognormally distributed, which we prove for a closely related probabilistic model. This study is motivated by the observation that the number of involutions in [n] is (n!)^(1/2) times a subexponential factor; more generally the number of permutations with all cycle lengths in a finite set S is n!^(1-1/m) times a subexponential factor, and the typical number of k-cycles is nearly n^(k/m)/k. Connections to pattern avoidance in involutions are also considered.
A signed graph $Gamma(G)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $Gamma(G)$. The energy of a signed graph $Gamma(G)$ is the sum of the absolute values of the eigenvalues of the adjacency matrix $A(Gamma(G))$ of $Gamma(G)$. The random signed graph model $mathcal{G}_n(p, q)$ is defined as follows: Let $p, q ge 0$ be fixed, $0 le p+q le 1$. Given a set of $n$ vertices, between each pair of distinct vertices there is either a positive edge with probability $p$ or a negative edge with probability $q$, or else there is no edge with probability $1-(p+ q)$. The edges between different pairs of vertices are chosen independently. In this paper, we obtain an exact estimate of energy for almost all signed graphs. Furthermore, we establish lower and upper bounds to the energy of random multipartite signed graphs.
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