Can gene-gene interactions result in gene expression?
I don't know anything about your algorithms. But i will try to explain your the format of data that was given to you. I don't know your background in Biology, so i will assume it is not your expertise field and will make some simplifications of the subject.  

What is gene expression?
To understand what is a gene expression, you need to understand what is a gene: 


  A gene is a sequence in the DNA (composed of 4 bases A,T,C and G) that can be transcribed by a protein, in your context we will say it always start with a start codon (a codon is a triplet of DNA bases) and stop with a stop codon. It is usually around a thousand bases long. The transcription will give you a RNA, and that RNA can be translated (note the difference with transcribed) into a whole new protein.  


Now the gene Expression is a measure of the amount of RNA from the gene you are looking for. In a cell you can have around 10000-100000 copies of that RNA; the raw count is not really stable as you can have extract two cells or three and it will change your "expression". Most of the time we normalize the count by the count of a bunch of known genes called housekeeping genes. The particularity of these genes is that their expression is quite stable.


  The figures you have are a ratio between RNA copies of your gene of interest and RNA copies of a stable gene (in term of expression).  


What does G1(-/-) and wt mean?

Usually when we (biologist) call a sample wt it means wild type, in other words, a normal individual. G1(-/-) means that this sample is from an individual with the G1 gene inactive (could be obtained by genetic modification).

What is a gene-gene interaction?
An example of interaction first: let's imagine a gene coding for an enhancer (booster) of transcription. the presence of this gene allow another gene to be expressed in the cell, if this gene isn't present or isn't transcribed, the other gene won't be expressed either.

How can we see interaction there?

Let's look at line one, the control (normal individual), We see that G4 (column) is almost unexpressed (0.07). In normal conditions, this gene is not expressed. Now let's look at mutant G1(-/-) and G6(-/-), we see the expression of G4 is rising (0.37;0.23) compared to the control (wt).
We can suppose than G1 and G6 are acting as inhibitors of the G4 gene.