The first cycles in an evolving graph
Discrete Mathematics
Generating functionology
A calculus for the random generation of labelled combinatorial structures
Theoretical Computer Science
An introduction to the analysis of algorithms
An introduction to the analysis of algorithms
Open problems of Paul Erd&ohuml;s in graph theory
Journal of Graph Theory
The asymptotic number of labeled graphs with n vertices, q edges, and no isolated vertices
Journal of Combinatorial Theory Series A
The enumeration of labeled graphs by number of cutpoints
Discrete Mathematics
Some Remarks on Sparsely Connected Isomorphism-Free Labeled Graphs
LATIN '00 Proceedings of the 4th Latin American Symposium on Theoretical Informatics
Combinatorial Enumeration
Analytic Combinatorics
Random 2-XORSAT at the satisfiability threshold
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
| Hi-index | 5.23 |
Given a set ξ = {H1,H2,...} of connected non-acyclic graphs, a ξ-free graph is one which does not contain any member of ξ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let Wk,ξ be the exponential generating function (EGF for brief) of connected ξ-free graphs of excess equal to k (k ≥ 1). For each fixed ξ, a fundamental differential recurrence satisfied by the EGFs Wk,ξ is derived. We give methods on how to solve this nonlinear recurrence for the first few values of k by means of graph surgery. We also show that for any finite collection ξ of non-acyclic graphs, the EGFs Wk,ξ are always rational functions of the generating function, T, of Cayley's rooted (non-planar) labelled trees. From this, we prove that almost all connected graphs with n nodes and n + k edges are ξ-free, whenever k = o(n1/3) and |ξ| n nodes has approximately n/2 edges. In particular, the probability distribution that it consists of trees, unicyclic components, ..., (q + 1)-cyclic components all ξ-free is derived. Similar results are also obtained for multigraphs, which are graphs where self-loops and multiple-edges are allowed.