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 the incidence-signed graph 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 infinitely many \((i+1)\) -path connected simplicial complexes K with \(B_i(K)\) being balanced.
