The largest normalized Laplacian eigenvalue and incidence balancedness of simplicial complexes

Let K be a simplicial complex, and let Δᵢᵘᵖ(K) be the i-th up normalized Laplacian of K. Horak and Jost showed that the largest eigenvalue of Δᵢᵘᵖ(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 Δᵢᵘᵖ(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 Δᵢᵘᵖ(K), which generalizes the corresponding result in graph case. Finally we gave some classes of infinitely many simplicial complexes K with Δᵢᵘᵖ(K) having an eigenvalue $i+2$ by using wedge, Cartesian product and duplication of motifs.