The smallest normalized signless $\infty$-Laplacian eigenvalue for non-bipartite connected graphs

TL;DR

This paper investigates the smallest normalized signless infinity-Laplacian eigenvalue for non-bipartite connected graphs, establishing a relationship with the minimal infinity-norm of generalized inverses of the weighted signless incidence matrix.

Contribution

It introduces a novel characterization of the smallest eigenvalue as the reciprocal of the minimal infinity-norm of certain generalized inverses, extending spectral graph theory.

Findings

Established the equality of the eigenvalue with the reciprocal norm

Provided an illustrative example demonstrating the theoretical result

Extended understanding of signless Laplacian eigenvalues in non-bipartite graphs

Abstract

In this paper, we aim to study the smallest normalized signless $\infty$-Laplacian eigenvalue $\mu_{\infty}$, a generalisation of the smallest signless Laplacian eigenvalue. For a non-bipartite connected graph, we show that the invariant $\mu_{\infty}$ equals to the reciprocal of the minimal $\infty$-norm of the generalized inverses of the weighted signless incidence matrix. An example is also given to illustrate the result.