The largest Laplacian eigenvalue and the balancedness of simplicial complexes

Let $K$ be a simplical complex, and let $\mathcal{L}_i^{up}(K), \mathcal{Q}_i^{up}(K)$ be the $i$-th up Laplacian and signless Laplacian of $K$, respectively. In this paper we proved that the largest eigenvalue of $\mathcal{L}_i^{up}(K)$ is not greater than the largest eigenvalue of $\mathcal{Q}_i^{up}(K)$; furthermore, if $K$ is $(i+1)$-path connected, then the equality holds if and only if the $i$-th incidence signed graph $B_i(K)$ of $K$ is balanced. As an application we provided an upper bound for the largest eigenvalue of the $i$-th up Laplacian of $K$, which improves the bound given by Horak and Jost and generalizes the result of Anderson and Morley on graphs.We characterized the balancedness of simplicial complexes under operations such as wedge sum, join, Cartesian product and duplication of motifs. For each $i \ge 0$, by using wedge sum or duplication of motifs, we can construct an infinitely many $(i+1)$-path connected simplicial complexes $K$ with $B_i(K)$ being balanced.