303 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
On a conjecture about tricyclic graphs with maximal energy
For a given simple graph , the energy of , denoted by , is defined as the sum of the absolute values of all eigenvalues of its
adjacency matrix, which was defined by I. Gutman. The problem on determining
the maximal energy tends to be complicated for a given class of graphs. There
are many approaches on the maximal energy of trees, unicyclic graphs and
bicyclic graphs, respectively. Let denote the graph with vertices obtained from three copies of and a path by
adding a single edge between each of two copies of to one endpoint of the
path and a single edge from the third to the other endpoint of the
. Very recently, Aouchiche et al. [M. Aouchiche, G. Caporossi, P.
Hansen, Open problems on graph eigenvalues studied with AutoGraphiX, {\it
Europ. J. Comput. Optim.} {\bf 1}(2013), 181--199] put forward the following
conjecture: Let be a tricyclic graphs on vertices with or
, then with equality
if and only if . Let denote the set of all
connected bipartite tricyclic graphs on vertices with three vertex-disjoint
cycles , and , where . In this paper, we try to
prove that the conjecture is true for graphs in the class ,
but as a consequence we can only show that this is true for most of the graphs
in the class except for 9 families of such graphs.Comment: 32 pages, 12 figure
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
- …