Asymptotic Properties of Labeled Connected Graphs

Asymptotic aspects of Cayley graphs

Arising from complete Cayley graphs Γn of odd cyclic groups Zn, an asymptotic approach is presented on connected labeled graphs whose vertices are labeled via equallymulticolored copies of K4 in Γn with adjacency of any two such vertices whenever they are represented by copies of K4 in Γn sharing two equally-multicolored triangles. In fact, these connected labeled graphs are shown to form a fam...

Asymptotic Number of General Cubic Graphs with given Connectivity

Let g(2n, l, d) be the number of general cubic graphs on 2n labeled vertices with l loops and d double edges. We use inclusion and exclusion with two types of properties to determine the asymptotic behavior of g(2n, l, d) and hence that of g(2n), the total number of general cubic graphs of order 2n. We show that almost all general cubic graphs are connected. Moreover, we determined the asymptot...

The Number of Labeled 2-Connected Planar Graphs

We derive the asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with n vertices contain many copies of any fixed planar graph, and this implies that almost all such graphs have large automorphism groups. ∗Research supported by the Australian Research Council and NSERC †Research supported by the ...

The enumeration of labeled graphs by number of cutpoints

A generating function is developed to express the number of labeled graphs with a fixed number of points and cutpoints in terms of the generating function of the number of blocks. An asymptotic bound is derived for the number of connected graphs with any number of cutpoints.

Preliminaries on the Asymptotic Number of H-free Labeled Connected Graphs with a given Number of Vertices and Edges