For
n≥2, define the graph
Gn:
- Vertices: Ordered pairs (x,y) with 1≤x,y≤n and x=y.
- Edges: (x1,y1)∼(x2,y2) if x1+y1=x2+y2 or x1−y1=x2−y2.
Define
L(Gn), the line graph of
Gn:
- Vertices: Edges of Gn.
- Edges: Two vertices are adjacent if their corresponding edges share a vertex in Gn.
Find the number of induced subgraphs of
L(Gn) where every vertex has degree 2.
Express this count as an explicit function of
n.