**I'VE COMPLETELY REVISED MY QUESTION** 


I wish to take a simple undirected graph (i.e. the complete graph K\_4)
Arbitrarily direct said graph, and then create a line graph from the directed version of the graph.


However, in Sage it appears to create a line graph that shows a connection between two edges (that are just inverses of each other), so what I really want is a line graph that doesn't give an edge connected to its own inverse.


That's why I asked if we could remove cycles of length 2, but that doesn't seem to solve the problem.


Here's what I am trying to work out:


G = graphs.RandomGNP(4,1)  

GD = G.to\_directed() #orients G  

m = GD.size() #number of edges of digraphG  

LG = GD.line\_graph() #the line graph of the digraph  

IM = identity\_matrix(QQ,GD.size())  

T = LG.adjacency\_matrix() #returns the adjacency matrix of the line graph  

var('u') #defines u as a variable  

X=IM-u\*T #defines a new matrix X  

Z=X.det() #defines polynomial in u aka **inverse of the Ihara zeta function**  

Z #computes determinant of X  

Z.coefficients(u) #extracts coefficients 


considering my graph is a complete graph on 4 vertices - the coefficients should be as such:  

[coeff,degree of u]  

[1,0], [0,1], [0,2],[-8,3],[-2,4]


**NOTE:**  

im only interested in coefficients up to the order of n=#of nodes in the graph, so here for K\_4 obviously n=4.  

where the coefficient of u^3 corresponds to the negative of twice the number of triangles in K\_4  

where the coefficient u^4 corresponds to the negative of twice the number of squares in K\_4



Ihara zeta function (graph theory) coefficients using a line graph
I received an answer (from fidelbc on sagemath) elsewhere and this is what I had wanted:



```
G = graphs.CompleteGraph(4)
D = G.to_directed()
L = D.line_graph()
L.delete_edges([((x,y,None), (y,x,None)) for x,y in G.edges( labels=None ) ])
L.delete_edges([((x,y,None), (y,x,None)) for y,x in G.edges( labels=None ) ])

```

