Let K be a simplicial complex, and let \(\Delta _i^{up}(K)\) be the i-th up normalized Laplacian of K. Horak and Jost showed that the largest eigenvalue of \(\Delta _i^{up}(K)\) is at most \(i+2\) , and characterized the equality case by orientable or non-orientable circuits. In this paper, we establish a relationship between the eigenvalue \(i+2\) and the balancedness of the signed incidence graph of K, namely, \(\Delta _i^{up}(K)\) has an eigenvalue \(i+2\) if and only if K has an \((i+1)\) -path connected component \(K'\) whose i-th signed incidence graph \(B_i(K')\) is balanced. We also characterize the multiplicity of \(i+2\) as an eigenvalue of \(\Delta _i^{up}(K)\) , and construct infinitely many simplicial complexes K with \(\Delta _i^{up}(K)\) having an eigenvalue \(i+2\) by using wedges, Cartesian products, or duplication of motifs.
