On the Laplacian coefficients of Signed Graphs

Let Γ = (G, σ) be a signed graph, where G is its underlying graph and σ its sign function (defined on edges of G). A signed graph Γ', the subgraph of Γ, is its signed TU-subgraph if the signed graph induced by the vertices of Γ' consists of trees and/or unbalanced unicyclic signed graphs. Let L(Γ) = D(G) − A(Γ) be the Laplacian of Γ. In this paper we express the coefficient of the Laplacian characteristic polynomial of Γ based on the signed TU-subgraphs of Γ, and establish the relation between the Laplacian characteristic polynomial of a signed graph with adjacency characteristic polynomials of its signed line graph and signed subdivision graph. As an application, we identify the signed unicyclic graphs having extremal coefficients of the Laplacian characteristic polynomial.