On the Laplacian coefficients of signed graphs

Abstract

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 graphΓ′, the subgraph of Γ, is its signed TU-subgraph if the signed graph induced by the vertices ofΓ′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

Spectral characterizations of signed lollipop graphs

“…Now, we recall some useful formulas, given in [5], which relate the Laplacian polynomial of a signed graph to the adjacency polynomials of its opportunely defined signed subdivision graph and signed line graph. In order to do so, we need to introduce a special oriented vertex-edge incidence matrix B η of a signed graph Γ = (G, σ) with n vertices and m edges.…”

“…Hence, S(Γ η ) and L(Γ η ) are uniquely defined up to switching isomorphisms, and for this reason the index η will be not anymore specified. For further details, the interested reader is referred to [5].…”

“…For a (possibly) complete bibliography on signed graphs, the reader is referred to [22]. For all notations not given here the reader is referred to [23] for signed graphs, to [8] for graph spectra, and to [5,16] for some basic results on the Laplacian spectral theory of signed graphs.…”

“…Recently in [5] the authors have considered an expression for the coefficient of the L-polynomial of a signed graph and they gave some formulas which express the L-polynomial by means of the A-polynomial of some related signed graphs. In this article, we consider those results to define some spectral invariants and study their effect on the combinatorial structure of the signed graphs.…”

A note on the least (normalized) laplacian eighva;ue of signed graphs

“…In [2], some parameters called frustration number and frustration index measuring how far a signed graph is to be balanced have been investigated in terms of the least eigenvalue of Laplacian of signed graph. More results on the spectra of signed graphs can be found in [1,2,3,6,11,12,16,13,14,5,7,8].…”