If I understand your question correctly...
First delete all edges that match your condition.
Then delete all vertices with zero neighbours.
Reproducible example with random data, where I don't want to plot correlations less than 0.1:
set.seed(999);mydata=matrix(runif(24),ncol=2)
rownames(mydata)=LETTERS[1:12]
g=graph.adjacency(cov(t(mydata)),weighted=TRUE)
plot(g)
that's a complete graph of 12 vertices.
g=delete.edges(g, which(E(g)$weight <=.1)) # here's my condition.
plot(g)
that leaves a skinny graph with a few detached vertices.
g=delete.vertices(g,which(degree(g)<1))
plot(g)
that cleans up.