Maximum clustering coefficient of graphs with given number of vertices and edges

The Asymptotic Number of Labeled Connected Graphs with a Given Number of Vertices and Edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q 5 N = ( ) edges. Let x = q/n and k = q n. We determine functions w k 1 , a(x ) and cp(x) such that c (n , q) w k ( z ) e n r p ( x ) e o ( x ) uniformly for all n and q 2 n. If Q > O is fixed, n+= and 4q > ( 1 + ~ ) n log n, this formula simplifies to c(n, q) ( t ) exp(-ne-zq’n). On the other hand, if k = o(n”’), this f...

The maximum number of Hamiltonian cycles in graphs with a fixed number of vertices and edges

The problem studied in this paper is that of nding the maximum number of Hamiltonian cycles in a graph with a given number of vertices and edges. The main results are a lower bound and an upper bound, both given by closedform formulas, for the maximum number of Hamiltonian cycles in a graph with a given number of vertices and edges. c © 2000 Elsevier Science B.V. All rights reserved.

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

The asymptotic number of labelled weakly-connected digraphs with a given number of vertices and edges

Edward A. Benderl Depariment of Mathel1wtics, University of California, San Diego, La Jolla, CA 92093, USA E. Rodney Canfieldl Department of Computer Science, University of Georgia, Athens, GA 30602, USA Brendan D. McKay2 Computer Science Department, Australian National University, GPO Box 4, ACT 2601., Australia In a recent paper, we determined the asymptotic number of labelled connected graph...

The Maximum Number of Edges in a Strongly Multiplicative Graph