944 research outputs found
Spanning trees and even integer eigenvalues of graphs
For a graph , let and be the Laplacian and signless
Laplacian matrices of , respectively, and be the number of
spanning trees of . We prove that if has an odd number of vertices and
is not divisible by , then (i) has no even integer
eigenvalue, (ii) has no integer eigenvalue , and
(iii) has at most one eigenvalue and such an
eigenvalue is simple. As a consequence, we extend previous results by Gutman
and Sciriha and by Bapat on the nullity of adjacency matrices of the line
graphs. We also show that if with odd, then the multiplicity
of any even integer eigenvalue of is at most . Among other things,
we prove that if or has an even integer eigenvalue of
multiplicity at least , then is divisible by . As a very
special case of this result, a conjecture by Zhou et al. [On the nullity of
connected graphs with least eigenvalue at least , Appl. Anal. Discrete
Math. 7 (2013), 250--261] on the nullity of adjacency matrices of the line
graphs of unicyclic graphs follows.Comment: Final version. To appear in Discrete Mat
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 triangle-free graphs with rank 6
The rank of a graph G is defined to be the rank of its adjacency matrix A(G).
In this paper we characterize all connected triangle-free graphs with rank 6
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
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