Invariants of graphs and hypergraphs

Case 1. The shortest (u, i )-walk in graph G and hypergraph H consists only edges of degree two. In this case paths have the same length as G and H describe the same molecular structure F on the same set of vertices and as (u, u)-walks are the shortest. Thus 

Case 2. The shortest u, u)G-walk in graph G consists only edges of degree two, and the shortest (u, u)jy-walk in hypergraph H contains at least one edge E of degree more than two. 

Suppose that the same set of vertices belongs to both paths  

As degE 2, the distance between all vertices belonging to the same edge is equal to one and 

Now suppose that (it, u)-paths contain different sets of vertices  

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