Interlacing Properties of Eigenvalues of Laplacian and Net-Laplacian Matrix of Signed Graphs

This paper explores interlacing inequalities in the Laplacian spectrum of signed cycles and investigates interlacing relationship between the spectrum of the net-Laplacian of a signed graph and its subgraph formed by removing a vertex together with its incident edges. Additionally, an inequality is derived between the net-Laplacian spectrum of a complete co-regular signed graph $\mathrm{\Gamma }$ and the Laplacian spectrum of the graph obtained by removing any vertex $v$ from $\mathrm{\Gamma }$. Also for a signed graph $\mathrm{\Gamma }$, the net-Laplacian matrix is normalized and an inequality is derived between the spectrum of the normalized net-Laplacian of a signed graph and its subgraph, formed by contraction of edge and vertex.