Spectra and balance of signed graphs obtained from vertex splittings

This paper introduces new constructions of nonregular cospectral signed graphs via two operations: the neighbours splitting (NS) join and the non-neighbours splitting (NNS) join. We compute the adjacency and Laplacian characteristic polynomials for each join, allowing spectral analysis for arbitrary signed graphs and explicit eigenvalue calculations for co-regular signed graphs. A second approach employs pseudo-potential functions to define more robust switching-stable versions of these joins, preserving spectral properties under switching equivalence. As an application of these techniques, we construct infinite families of nonisomorphic signed graphs exhibiting cospectrality for both adjacency and Laplacian matrices. We also characterise balancedness conditions for each of the constructions.