Natural Science Forum / Physics / General Physics / October 2007

The concept of Energy has been applied to graphs for quite a while,
and a paper by Dragos Cvetkovic and Jason Grout (Grant?), "Graphs with
extremal energy should have a small number of distinct eigenvalues,"
arXiv: 0710.5664 v1 [math.CO] 30 )ct 2007, 17 pages, U. Belgrade
Servia and Brigham Young U. USA respectively, shows that it has
several features of Probable Causation/Influence (PI) even though the
authors don't know it.  These include additivity (which with
subtractivity is a key characteristic of PI) and simplicity, which in
their paper takes the form of "small number of distinct
eigenvalues" (2 in the best case scenario).

They define energy E of a graph G by:

1) E(G) = sum /xi/, i in set I = {1, 2, ..., n}

where xi are eigenvalues of G and G is a simple undirected graph on n
vertices and m > 0 edges.  This definition comes from I. Gutman
(1978).  Other researchers define the energy as the sum of /xi(A(G))/
where A(G) is the adjacency matrix of graph J, and it turns out that
these absolute values equal singular values of A(G).

It is well known that sum xi^2 = 2m, i in set I, and several key 1/2
factor bounds are known such as:

2) E(G) < = (n/2)(1 + n^(1/2)) for graph G on n vertices

An almost reverse inequality using > = except that the factor (1 -
epsilon) is inserted on the right hand side has been confirmed in the
sense that for every epsilon > 0 and almost all n, there exists a
graph G on n vertices such that the almost reverse inequality holds.
No "there exists" is required in (2), however.

I won't state or prove their main theorem, Theorem 1, because it
involves a slightly lengthy procedure involving Lagrange multipliers,
but interestingly the factors 2 and 1/2 appear in several equations in
this procedure.

Osher Doctorow