Wright's formulae for the number of connected sparsely edged graphs

1983

Glasgow Math. J.

An (n, q) graph is a graph on n labelled points and q lines, no loops and no multiple lines. We write N = ½n(n – 1), B(a, b) = a!/{b!(a – b)!} and B(a, 0) = 1, so that there are just B(N, q)different (n, q) graphs. Again h(n, q) is the number of Hamiltonian (n, q) graphs. Much attention has been devoted to the problem of determining for which q = q(n) “almost all” (n, q) graphs are Hamiltonian, i.e. for which q we haveas n → ∞. I proved [8, Theorem 4] that qn–3/2; → ∞ is a sufficient condition by showing that, for such q, almost all (n, q) graphs have about the average number of Hamiltonian circuits (H.c.s).

Section: T H E N U M B E R O F Sparsely E D G E D L a B E L L E D H Amentioning

confidence: 99%

“…We find that T 23 = |n(n-4)(2n-7), T 33 = §n(n-3), T 43 = n and T s3 = 0 for ss*5. Hence, by (4) and 8 With sufficient labour, we could use this method to calculate h(n, n + 4), but the prospect is somewhat daunting. Beyond k = 4, the method seems impracticable.…”

mentioning

confidence: 99%

The number of sparsely edged labelled Hamiltonian graphs

1983

Glasgow Math. J.

“…If the special line belongs to a suspended path and if its removal disconnects the graph, we put the graph in set 98 3 , In each of the remaining graphs, the removal of the special line leaves the graph connected; if the suspended path is of length one, i.e. consists of the special line alone, we put the graph in sub-set 58 4 , if not, in sub-set 98 5 .…”

Section: Direct Combinatorial Proof Of Theoremmentioning

confidence: 99%

“…where the c ks are those given in Theorem 4 of [12] and b k =c k -3 k , c k = -c M _ 3 k . These can be calculated by computer by the methods described in [5] and [12] and, as shown in [14], b k = 3 k 2~k(fc -\)\d k , where d k is the sequence described in Theorem 5 below.…”

Section: Sparsely-edged Smooth Graphsmentioning

confidence: 99%

Enumeration of smooth labelled graphs

Proceedings of the Royal Society of Edinburgh: Section A Mathem

1982

SynopsisAn (n, q) graph is a graph on n labelled points and q lines without loops or multiple lines. We write ν(n, q) for the number of smooth (n, q) graphs, i.e. connected graphs without end points, and ν = V(Z, Y) = ∑n,q ν(n,q)ZnYq /n! for the exponential generating function of ν(n,q). We use the Riddell “core and mantle” method to find an explicit form for V (not, as usual with this method, only a functional equation). From this we deduce a partial differential equation satisfied by V. We interpret this equation in purely combinatorial terms. We write Vk = ∑ n ν(n, n + k)Xn/n! and find a recurrence formula for Vk for successive k. We use these and other results to find an asymptotic expansion for ν(n,q) as n→∞ when (q/n) − log n − log log n→ + ∞ and an asymptotic approximation to ν(n,n + k) when 0 < k = o and to log ν(n, n + k) when k < (1−ε).

The number of connected sparsely edged graphs. III. Asymptotic results

Journal of Graph Theory

1980

The number of connected graphs on n labeled points and 9 lines (no loops, no multiple lines) is f(n.9). In the first paper of this series I showed how to find an (increasingly complicated) exact formula for f(n,n+k) for general n and successive k . The method would give an asymptotic approximation t o f(n,n+k) for any fixed k as nw. Here I find this approximation when k = ~( n l '~) , a much more difficult matter. The problem of finding an approximation t o f(n.9) when 9 > n + Cn"3 and ( 2 q l n )log n --w is open.