A signed graph is a pair Γ = (G; ), where x = (V (G);E(G)) is a graph and
E(G) -> {+1;−1} is the sign function on the edges of G. For any > [0; 1] we consider the
matrix
Aα(Γ) = αD(G) + (1 −α )A(Γ);
where D(G) is the diagonal matrix of the vertex degrees of G, and A(Γ) is the adjacency
matrix of Γ. Let mAα(Γ) be the multiplicity of α as an A(Γ)-eigenvalue, and let G have
p(G) pendant vertices, q(G) quasi-pendant vertices, and no components isomorphic to K2. It
is proved that
mA(Γ)() = p(G) − q(G)
whenever all internal vertices are quasi-pendant. If this is not the case, it turns out that
mA(Γ)() = p(G) − q(G) +mN(Γ)();
where mN(Γ)() denotes the multiplicity of as an eigenvalue of the matrix N(Γ) obtained
from A(Γ) taking the entries corresponding to the internal vertices which are not quasipendant.
Such results allow to state a formula for the multiplicity of 1 as an eigenvalue of
the Laplacian matrix L(Γ) = D(G) − A(Γ). Furthermore, it is detected a class G of signed
graphs whose nullity – i.e. the multiplicity of 0 as an A(Γ)-eigenvalue – does not depend on the
chosen signature. The class G contains, among others, all signed trees and all signed lollipop
graphs. It also turns out that for signed graphs belonging to a subclass G ` G the multiplicity
of 1 as Laplacian eigenvalue does not depend on the chosen signatures. Such subclass contains
trees and circular caterpillars