47 research outputs found
Further results on the nullity of signed graphs
The nullity of a graph is the multiplicity of the eigenvalues zero in its
spectrum. A signed graph is a graph with a sign attached to each of its edges.
In this paper, we obtain the coefficient theorem of the characteristic
polynomial of a signed graph, give two formulae on the nullity of signed graphs
with cut-points. As applications of the above results, we investigate the
nullity of the bicyclic signed graph , obtain the
nullity set of unbalanced bicyclic signed graphs, and thus determine the
nullity set of bicyclic signed graphs.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1207.6765,
arXiv:1107.0400 by other author
The extremal problems on the inertia of weighted bicyclic graphs
Let be a weighted graph. The number of the positive, negative and zero
eigenvalues in the spectrum of are called positive inertia index,
negative inertia index and nullity of , and denoted by ,
, , respectively. In this paper, sharp lower bound on
the positive (resp. negative) inertia index of weighted bicyclic graphs of
order with pendant vertices is obtained. Moreover, all the weighted
bicyclic graphs of order with at most two positive, two negative and at
least zero eigenvalues are identified, respectively.Comment: 12 pages, 5 figures, 2 tables. arXiv admin note: text overlap with
arXiv:1307.0059 by other author
The Nullity of Bicyclic Signed Graphs
Let \Gamma be a signed graph and let A(\Gamma) be the adjacency matrix of
\Gamma. The nullity of \Gamma is the multiplicity of eigenvalue zero in the
spectrum of A(\Gamma). In this paper we characterize the signed graphs of order
n with nullity n-2 or n-3, and introduce a graph transformation which preserves
the nullity. As an application we determine the unbalanced bicyclic signed
graphs of order n with nullity n-3 or n-4, and signed bicyclic signed graphs
(including simple bicyclic graphs) of order n with nullity n-5
On the positive and negative inertia of weighted graphs
The number of the positive, negative and zero eigenvalues in the spectrum of
the (edge)-weighted graph are called positive inertia index, negative
inertia index and nullity of the weighted graph , and denoted by ,
, , respectively. In this paper, the positive and negative
inertia index of weighted trees, weighted unicyclic graphs and weighted
bicyclic graphs are discussed, the methods of calculating them are obtained.Comment: 12. arXiv admin note: text overlap with arXiv:1107.0400 by other
author
The inertia of weighted unicyclic graphs
Let be a weighted graph. The \textit{inertia} of is the triple
, where
are the number of the positive, negative and zero
eigenvalues of the adjacency matrix of including their
multiplicities, respectively. , is called the
\textit{positive, negative index of inertia} of , respectively. In this
paper we present a lower bound for the positive, negative index of weighted
unicyclic graphs of order with fixed girth and characterize all weighted
unicyclic graphs attaining this lower bound. Moreover, we characterize the
weighted unicyclic graphs of order with two positive, two negative and at
least zero eigenvalues, respectively.Comment: 23 pages, 8figure