Graphico-Topological Invariants

The simplest relations have to do with atom and bond counts. If V,(C) = V, denotes the number of vertices of degree i in graph G and E G) = E denotes the number of edges, then 

But there are some additional relations involving invariants of more fundamental graphico-topological content. 

for a graph G a simple invariant is the difference tV(G) - E G) between the vertex and edge counts for G. Clearly this (which is very nearly the cyclomatic number of G) is additive in terms of any disconnected components G might have. It takes the value 0 for a simple cycle and the value 1 for a double cycle such as naphthalene. 

Second, a fundamental topological invariant for a geometric complex K is the Euler characteristic xW- This may be defined to be a real-valued function invariant under homeomorphism such that 

Xdine segment) = 1 X(disk)= 1 X(cycle) = 0 xfsphere surface) = 2 Xftorus surface) = 0 X( 1-handled basket) = 0 Xfdouble torus surface) = -2 X(cylinder, without ends) = 0 

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The fundamental graphico-combinatorial result we are currently after relates the topological, graphical, and embedding invariants of the preceding three paragraphs. 

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