Sorry for taking so long to respond. I didn't get a notification when you posted your response. Hopefully you already figured this out. If not, here's an explanation.
If I understand you correctly, your saying that when you open data in cytoscape, it only displays nodes, not edges. Try making a sif file. Here's an example:
Copy and paste the following into a text file and save it as "test.sif" or something like that
gene1 edge gene2
gene1 edge gene3
gene2 edge gene3
gene4 edge gene5
gene4 edge gene6
gene4 edge gene7
When you open cytoscape (version 3.1.1 is the one I'm using), select "From Network file..." then select test.sif, and press ok a couple times. Then you should see a network with edges. The word "edge" is totally arbitrary. You would get the same result if you had for example:
gene1 pink gene2
gene1 green gene3
gene2 yellow gene3
gene4 nothing gene5
gene4 wedge gene6
gene4 pledge gene7
Distinguishing between different kinds of edges can be useful for some things, but it sounds like in your case it doesn't matter.
I think this question is unclear. Is it that you want your gene names to be the nodes, and you want to draw edges between genes that are somehow related (co-expressed, maybe)? Is your question about how to format input for Cytoscape, or how to determine which genes are related to each other?
If that it's about Cytoscape input, then I think the simplest way is just to generate a sif file, which is a tab separated file with three columns, the first is the source node, the second is the edge type (which can be arbitrary, in a lot of cases you only have one edge type in the network, so you don't need to distinguish them), the third is the destination node.
Another simple option is just to import network data from an excel file.
You can read more about Cytoscape file formats at the Cytoscape documentation.