Lattices with lots of congruence energy

Abstract

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$(A)|=2^{n-1}$.