Euler numbers and diametral paths in Fibonacci cubes, Lucas cubes and Alternate Lucas cubes

TL;DR

This paper investigates the number of diametral paths in Fibonacci, Lucas, and Alternate Lucas cubes, revealing their enumeration is connected to Euler numbers and alternating permutations.

Contribution

It provides bijective proofs linking diametral path counts in these cubes to Euler numbers, a novel combinatorial insight.

Findings

Number of diametral paths related to Euler numbers

Enumeration connected to alternating permutations

Bijective proofs established for these relationships

Abstract

The diameter of a graph is the maximum distance between pairs of vertices in the graph. A pair of vertices whose distance is equal to its diameter are called diametrically opposite vertices. The collection of shortest paths between diametrically opposite vertices are referred as diametral paths. In this work, we enumerate the number of diametral paths for Fibonacci cubes, Lucas cubes and Alternate Lucas cubes. We present bijective proofs that show that these numbers are related to alternating permutations and are enumerated by Euler numbers.