The largest normalized Laplacian eigenvalue and incidence balancedness of simplicial complexes

Let K be a simplicial complex, and let ᵢ^up (K) be the i-th up normalized Laplacian of K. Horak and Jost showed that the largest eigenvalue of ᵢ^up (K) is at most i+2, and characterized the equality case by the orientable or non-orientable circuits. In this paper, by using the balancedness of signed graphs, we show that ᵢ^up (K) has an eigenvalue i+2 if and only if K has an (i+1) -path connected component K' such that the i-th signed incidence graph Bᵢ (K') is balanced, which implies Horak and Jost's characterization. We also characterize the multiplicity of i+2 as an eigenvalue of ᵢ^up (K), which generalizes the corresponding result in graph case. Finally we gave some classes of infinitely many simplicial complexes K with ᵢ^up (K) having an eigenvalue i+2 by using wedge, Cartesian product and duplication of motifs.