25 research outputs found
On the Existence of Non-golden Signed Graphs
A signed graph is a pair Î = (G,Ï), where G = (V(G),E(G)) is a graph and Ï : E(G)â{+1,â1} is the sign function on the edges of G. For a signed graph we consider the least eigenvector λ(Î) of the Laplacian matrix defined as L(Î) = D(G)âA(Î), where D(G) is the matrix of vertices degrees of G and A(Î) is the signed adjacency matrix.
An unbalanced signed bicyclic graph is said to be golden if it is switching equivalent to a
graph Î satisfying the following property: there exists a cycle C in Î and a λ(Î)-eigenvector
x such that the unique negative edge pq of Î belongs to C and detects the minimum of the
set Sx(Î,C) = { |xrxs| | rs â E(C) }.
In this paper we show that non-golden bicyclic graphs with frustration index 1 exist for each
n â„ 5
Signed bicyclic graphs with minimal index
The index of a signed graph \Sigma = (G; \sigma) is just the largest eigenvalue
of its adjacency matrix. For any n > 4 we identify the signed graphs achieving the
minimum index in the class of signed bicyclic graphs with n vertices. Apart from the n = 4 case, such graphs are obtained by considering a starlike tree with four branches of suitable length (i.e. four distinct paths joined at their end vertex u) with two additional negative independent edges pairwise joining the four vertices adjacent to u. As a by-product, all signed bicyclic graphs containing a theta-graph and whose index is less than 2 are detected
On the multiplicity of α as an A_α (Î)-eigenvalue of signed graphs with pendant vertices
A signed graph is a pair Î = (G; ), where x = (V (G);E(G)) is a graph and
E(G) -> {+1;â1} is the sign function on the edges of G. For any > [0; 1] we consider the
matrix
Aα(Î) = αD(G) + (1 âα )A(Î);
where D(G) is the diagonal matrix of the vertex degrees of G, and A(Î) is the adjacency
matrix of Î. Let mAα(Î) be the multiplicity of α as an A(Î)-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(Î)() = p(G) â q(G)
whenever all internal vertices are quasi-pendant. If this is not the case, it turns out that
mA(Î)() = p(G) â q(G) +mN(Î)();
where mN(Î)() denotes the multiplicity of as an eigenvalue of the matrix N(Î) obtained
from A(Î) 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(Î) = D(G) â A(Î). Furthermore, it is detected a class G of signed
graphs whose nullity â i.e. the multiplicity of 0 as an A(Î)-eigenvalue â 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
Laplacian spectral properties of signed circular caterpillars
A circular caterpillar of girth n is a graph such that the removal of all pendant vertices yields a cycle Cn of order n. A signed graph is a pair Î = (G, Ï), where G is a simple graph and Ï â¶ E(G) â {+1, â1} is the sign function defined on the set E(G) of edges of G. The signed graph Î is said to be balanced if the number of negatively signed edges in each cycle is even, and it is said to be unbalanced otherwise. We determine some bounds for the first n Laplacian eigenvalues of any signed circular caterpillar. As an application, we prove that each signed spiked triangle (G(3; p, q, r), Ï), i. e. a signed circular caterpillar of girth 3 and degree sequence Ïp,q,r = (p + 2, q + 2, r + 2, 1,..., 1), is determined by its Laplacian spectrum up to switching isomorphism. Moreover, in the set of signed spiked triangles of order N, we identify the extremal graphs with respect to the Laplacian spectral radius and the first two Zagreb indices. It turns out that the unbalanced spiked triangle with degree sequence ÏNâ3,0,0 and the balanced spike triangle (G(3; p, ^ q, ^ r^), +), where each pair in {p, ^ q, ^ r^} differs at most by 1, respectively maximizes and minimizes the Laplacian spectral radius and both the Zagreb indices
Unbalanced signed graphs with extremal spectral radius or index
Let ËG = (G, Ï) be a signed graph, and let Ï( ËG ) (resp. λ1( ËG )) denote the spectral radius
(resp. the index) of the adjacency matrix A( ËG) . In this paper we detect the signed graphs
achieving the minimum spectral radius m(SRn), the maximum spectral radius M(SRn),
the minimum index m(In) and the maximum index M(In) in the set U_n of all unbalanced
connected signed graphs with n â„ 3 vertices. From the explicit computation of the four
extremal values it turns out that the difference m(SRn)âm(In) for n â„ 8 strictly increases
with n and tends to 1, whereas M(SRn) â M(In) strictly decreases and tends to 0
The minimal spectral radius with given independence number
In this paper, we determine the graphs which have the minimal spectral radius
among all the connected graphs of order and the independence number
Comment: 14 page
Line graphs of complex unit gain graphs with least eigenvalue -2
Let T be the multiplicative group of complex units, and let L(Ï) denote a line graph of a T-gain graph Ï. Similarly to what happens in the context of signed graphs, the real number min Spec(A(L(Ï)), that is, the smallest eigenvalue of the adjacency matrix of L(Ï), is not less than -2. The structural conditions on Ï ensuring that min Spec(A(L(Ï)) = -2 are identified. When such conditions are fulfilled, bases of the -2-eigenspace are constructed with the aid of the star complement technique
A lower bound for the first Zagreb index and its application
For a graph G, the first Zagreb index is defined as the sum of the squares of the vertices degrees. By investigating the connection between the first Zagreb index and the first three coefficients of the Laplacian characteristic polynomial, we give a lower bound for the first Zagreb index, and we determine all corresponding extremal graphs. By doing so, we generalize some known results, and, as an application, we use these results to study the Laplacian spectral determination of graphs with small first Zagreb index