  In 1978, motivated by E. H\"uckel's work in quantum chemistry, I. Gutman
introduced the concept of the energy of a finite simple graph $G$ as the sum of
the absolute values of the eigenvalues of the adjacency matrix of $G$. At the
time of writing, the MathSciNet search for "Title=(graph energy) AND Review
Text=(eigenvalue)" returns 351 publications, most of which going after Gutman's
definition.
  A congruence $\alpha$ of a finite algebra $A$ turns $A$ into a simple graph:
we connect $x\neq y\in A$ by an edge iff $(x,y)\in\alpha$; we let En$(\alpha)$
be the energy of this graph. We introduce the congruence energy CE$(A)$ of $A$
by CE$(A):=\sum\{$En$(\alpha): \alpha\in$ Con$(A)\}$. Let LAT$(n)$ and CDA$(n)$
stand for the class of $n$-element lattices and that of $n$-element congruence
distributive algebras of any type. For a class $\mathcal X$, let CE$(\mathcal
X):= \{$CE$(A): A\in \mathcal X\}$. We prove the following.
  (1) For $\alpha\in A$, En$(\alpha)/2$ is the height of $\alpha$ in the
equivalence lattice of $A$.
  (2) The largest number and the second largest number in CE(LAT($n$)) are
$(n-1)\cdot 2^{n-1}$ and, for $n\geq 4$, $(n-1)\cdot 2^{n-2}+2^{n-3}$; these
numbers are only witnessed by chains and lattices with exactly one two-element
antichain, respectively.
  (3) The largest number in CE(CDA($n$)) is also $(n-1)\cdot 2^{n-1}$, and if
CE$(A)=(n-1)\cdot 2^{n-1}$ for an $A\in$ CDA$(n)$, then Con$(A)$ is a boolean
lattice with size $|$Con$(A)|=2^{n-1}$.
