If we know the probability $P$ that there exists an edge between two vertices of an undirected graph, let’s say $P= 1/v$, where $v$ is the number of vertices in the graph, what is the probability that the graph has cycles? I’ve twisted my brain with this. Can anyone help?

A recent discussion, which may be found here, examined the problem of counting the number of acyclic digraphs on $n$ labelled nodes and having $k$ edges and indegree and outdegree at most one. It was established that the bivariate mixed generating function of this class $\mathcal{G}$ of graphs on $n$ nodes and with $k$ edges […]

I am trying to find polynomial, indicator function or sometimes called structure function to express whether a vertex-induced random subgraph $H$ of $G$ is connected or not. The polynomial $\phi(G’)$ should be the undirected version while I am trying to figure out the direction version. So where I cannot fully understand the second line: it […]

By a $G(n,p)$ graph we mean a graph on $n$ vertices, all possible edges are independently included randomly with probability $p$. What can be said about the number of connected components? For example, bounds or asymptotic behavior of the expected number of components as $n\rightarrow \infty$ or for $p$ close to 1.

By a random $n\times n$ bipartite graph, I mean a random bipartite graph on two vertex classes of size $n$, with the edges added independently, each with probability $p$. I want to find the probability that such a graph contains an isolated vertex. Let $X$ and $Y$ be the vertex classes. I can calculate the […]

Consider the set $V = \{1,2,\ldots,n\}$ and let $p$ be a real number with $0<p<1$. We construct a graph $G=(V,E)$ with vertex set $V$, whose edge set $E$ is determined by the following random process: Each unordered pair $\{i,j\}$ of vertices, where $i \neq j$, occurs as an edge in $E$ with probability $p$, independently […]

Trying to understand how the Voronoi Diagrams work I did a test generating the Voronoi diagram of the points obtained from The Chaos Game algorithm when it is applied to a $3$-gon. The result is a set of points part of the Sierpinski attractor, the Voronoi cells and the vertices of each cell. This is […]

Suppose a simple graph $G$. Now consider probability space $G(v;p)$ where $0\leq p\leq 1$ and $v$ vertices. I want to calculate globally-determined properties of $G(v;p)$ such as connectivity and expected value of indicator function $$\mathbb E_{p\sim [0,1]^n}(\phi(G))$$ in terms of st-connectedness where $p$ follows let say uniform distribution. I want to understand which area investigates […]

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