On the multiplicity of α as an Aα(Γ)-eigenvalue of signed graphs with pendant vertices

A signed graph is a pair Γ = ( G , σ ) , where x = ( V ( G ) , E ( G ) ) is a graph and σ : E ( G ) → { + 1 , − 1 } is the sign function on the edges of G . For any α ∈ [ 0 , 1 ] we consider the matrix A α ( Γ ) = α D ( 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 m A α ( Γ ) ( α ) be the multiplicity of α as an A α ( Γ ) -eigenvalue, and let G have p ( G ) pendant vertices, q ( G ) quasi-pendant vertices, and no components isomorphic to K 2 . It is proved that m A α ( Γ ) ( α ) = p ( G ) − q ( G ) whenever all internal vertices are quasi-pendant. If this is not the case, it turns out that m A α ( Γ ) ( α ) = p ( G ) − q ( G ) + m N α ( Γ ) ( α ) , where m N ( Γ ) ( α ) denotes the multiplicity of α as an eigenvalue of the matrix N α ( Γ ) obtained from A α ( Γ ) taking the entries corresponding to the internal vertices which are not quasi-pendant. 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.