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

Abstract

In this paper, we aim to study the smallest normalized signless $\infty$-Laplacian eigenvalue $\hat{I}{¼}_{\infty }$, a generalisation of the smallest signless Laplacian eigenvalue. For a non-bipartite connected graph, we show that the invariant $\hat{I}{¼}_{\infty }$ 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.