I am quite sure I have seen somewhere the connection between the characteristic polynomial of a (finite undirected) graph and its dual. I am not able to find it currently. Could you please refer me to the result? 


(eigenvalues would be enough)


By dual, I mean the graph where edges become vertices and adjacent if they intersect in a vertex. 
By characteristic polynomial I mean that of the adjacency matrix. 



Characteristic polynomial of the line graph (originally dual graph)
The term for the graph construction you are talking about is "line graph" (see [the Wikipedia article](http://en.wikipedia.org/wiki/Line_graph)). The eigenvalues of the adjacency matrix of the line graph $L(\Gamma)$ are closely related to the signless Laplacian eigenvalues of the original graph $\Gamma$, as explained for instance in section 1.4.5 of [these notes](http://homepages.cwi.nl/~aeb/math/ipm/ipm.pdf) (see also Chris Godsil and Gordon Royle's textbook on algebraic graph theory). An important spectral feature of line graphs is that their (adjacency matrix) eigenvalues are at least $-2$.


