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

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