unbalanced complete bipartite signed graphs ${k_{m, n}}^{sigma}$ having $m$ and $n$ as laplacian eigenvalues with maximum multiplicities

A signed graph $ {g}^{sigma} = (g, sigma) $ consists of an underlying graph $ g = (v, e) $ along with a signature function $ sigma: e rightarrow {-1, 1} $. a cycle in a signed graph is termed positive if it contains an even number of negative edges, and negative if it contains an odd number of negative edges. a signed graph is considered { balanced} if it has no negative cycles; otherwise, it is { unbalanced}. let $k_{m,n}$ be a { complete bipartite graph} on $m+n$ vertices. it is well known that for a balanced complete bipartite signed graph $ {k_{m,n}}^{sigma} $, the parameters $ m $ and $ n $ are laplacian eigenvalues with multiplicities $ n-1 $ and $ m-1 $, respectively. this raises a natural question about the maximum multiplicities of laplacian eigenvalues $ m $ and $ n $ in an unbalanced complete bipartite signed graph $ {k_{m,n}}^{sigma} $. in this paper, we demonstrate that the multiplicities of the laplacian eigenvalues $ m $ and $ n $ in an unbalanced complete bipartite signed graph $ {k_{m,n}}^{sigma} $ are at most $ n-2 $ and $ m-2 $, respectively. additionally, we characterize all the signed graphs for which $ m $ and $ n $ are laplacian eigenvalues with these maximum multiplicities.