203 research outputs found
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 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
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
On the minimum rank of not necessarily symmetric matrices : a preliminary study
The minimum rank of a directed graph G is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i, j) is an arc in G and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i _= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem
A note on the minimum skew rank of a graph
The minimum skew rank of a graph over a field
is the smallest possible rank among all skew symmetric matrices
over , whose (,)-entry (for ) is nonzero whenever
is an edge in and is zero otherwise. We give some new properties of
the minimum skew rank of a graph, including a characterization of the graphs
with cut vertices over the infinite field such that
, determination of the minimum skew rank of -paths
over a field , and an extending of an existing result to show that
for a connected graph
with no even cycles and a field , where is the matching
number of , and is the largest possible rank among
all skew symmetric matrices over
Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph
For a given graph G and an associated class of real symmetric matrices whose
off-diagonal entries are governed by the adjacencies in G, the collection of
all possible spectra for such matrices is considered. Building on the
pioneering work of Colin de Verdiere in connection with the Strong Arnold
Property, two extensions are devised that target a better understanding of all
possible spectra and their associated multiplicities. These new properties are
referred to as the Strong Spectral Property and the Strong Multiplicity
Property. Finally, these ideas are applied to the minimum number of distinct
eigenvalues associated with G, denoted by q(G). The graphs for which q(G) is at
least the number of vertices of G less one are characterized.Comment: 26 pages; corrected statement of Theorem 3.5 (a
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