The $\mu$-permanent, a new graph labeling, and a known integer sequence

TL;DR

This paper introduces the $$-permanent, a new graph labeling for trees based on a polynomial involving permutation inversions, and connects it to a known integer sequence through path labelings.

Contribution

It proposes a novel graph labeling method related to the $$-permanent and links it to an established integer sequence, expanding the understanding of graph invariants.

Findings

Defined the $$-permanent for matrices and related it to graph labelings.

Established a connection between path labelings and a known integer sequence.

Provided multiple examples illustrating the new labeling and its properties.

Abstract

Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $\mu$, we define the polynomial $$P_\mu(A)=\sum_{\sigma\in S_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)}\,\mu^{\ell(\sigma)}\; ,$$ as the $\mu$-permanent of $A$, where $\ell(\sigma)$ is the number of inversions of the permutation $\sigma$ in the symmetric group $S_n$. In this note, motivated by this notion, we discuss a new graph labeling for trees whose matrices satisfy certain $\mu$-permanental identities. We relate the number of labelings of a path with a known integer sequence. Several examples are provided.