In 1978, motivated by E. Hückel'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 α of a finite algebra A turns A into a simple graph: we connect x = y ∈ A by an edge iff (x, y) ∈ α; we let En(α) be the energy of this graph. We introduce the congruence energy CE(A) of A by CE(A) := {En(α) : α ∈ 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 X , let CE(X ) := {CE(A) : A ∈ X }. We prove the following. (1) For α ∈ A, En(α)/2 is the height of α in the equivalence lattice of A. (2) The largest number and the second largest number in CE(LAT(n)) are (n − 1) • 2 n−1 and, for n ≥ 4, (n − 1) • 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 ∠ . http://www.math.u-szeged.hu/~czedli/ . 2020 Mathematics Subject Classification. Primary 06B10; secondary 05C50 and 08B10 .
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