Lattices with lots of congruence energy

TL;DR

This paper introduces the concept of congruence energy for finite algebras, especially lattices, and analyzes its maximum values and structural implications, revealing connections between algebraic properties and graph energy.

Contribution

It defines the congruence energy of finite algebras and determines the maximum values for lattices and congruence distributive algebras, linking these maxima to specific lattice structures.

Findings

Maximum congruence energy for n-element lattices is (n-1)*2^{n-1}.

Second maximum occurs in lattices with one two-element antichain.

Maximum congruence energy for n-element congruence distributive algebras is also (n-1)*2^{n-1}, with specific structural characterization.

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