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In connection with Le Monde puzzle #46, I eventually managed to write an R program that generates graphs with a given number n of nodes and a given number k of edges leaving each of those nodes. (My early attempt was simply too myopic to achieve any level of success when n was larger than 10!) Here is the core of the R code:

A=42 #number of nodes
L=13 #number of edges
ApL=A+L
if ((A*L)%%2==1){
print("impossible graph")
}else{
con=matrix(0,A,A)
diag(con)=A #eliminate self-connection
suma=apply(con,1,sum)-A
while (min(suma)ApL-1)) vali=NULL
}else{
vali=slots[apply(con[slots,],1,sum)

and it uses a sort of annealed backward step to avoid simulating a complete new collection of neighbours when reaching culs-de-sac….

aclrtr=function(con,L){
#removes a random number of links among the nodes with L links
A=dim(con)[1]
ApL=A+L
while (max(apply(con,1,sum))==ApL){
don=sample(1:(L-1),1)
if (sum(apply(con,1,sum)==ApL)==1){
i=(1:A)[apply(con,1,sum)==ApL]
}else{
i=sample((1:A)[apply(con,1,sum)==ApL],1)
}
off=sample((1:A)[con[i,]==1],don)
con[i,off]=0
con[off,i]=0
}
con
}

There is nothing fancy or optimised about this code so I figure there are much better versions to be found elsewhere…

Ps-As noticed before, sample does not work on a set of length one, which is a bug in my opinion…. Instead, sample(4.5,1) returns a random permutation of (1,2,3,4).