# A random graph

**Discrete Mathematics**Level 5

A graph is constructed iteratively according to the following algorithm. The graph starts as a single vertex. With probability \(p\), the graph stops here. Otherwise, 3 new vertices are constructed and joined to this vertex. If we have a graph with more than one vertex, for each vertex created at the previous stage, 3 new vertices are construed and joined to it with probability \(1 - p.\) We repeat this process until no new vertices are added. When the expected number of vertices in this graph is \(100,\) the value of \(p\) can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b?\)