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Distance and distance signless Laplacian spread of connected graphs
For a connected graph on vertices, recall that the distance signless
Laplacian matrix of is defined to be ,
where is the distance matrix, and is the row sum of corresponding to vertex
. Denote by the
largest eigenvalue and the least eigenvalue of , respectively.
And denote by , the largest
eigenvalue and the least eigenvalue of , respectively. The
distance spread of a graph is defined as
, and
the distance signless Laplacian spread of a graph is defined as
. In this
paper, we point out an error in the result of Theorem 2.4 in "Distance spectral
spread of a graph" [G.L. Yu, et al, Discrete Applied Mathematics. 160 (2012)
2474--2478] and rectify it. As well, we obtain some lower bounds on ddistance
signless Laplacian spread of a graph
The distinguishing index of graphs with at least one cycle is not more than its distinguishing number
The distinguishing number (index) () of a graph is the
least integer such that has an vertex (edge) labeling with labels
that is preserved only by the trivial automorphism. It is known that for every
graph we have . The complete characterization of
finite trees with has been given recently.
In this note we show that if is a finite connected graph with at least
one cycle, then . Finally, we characterize all connected graphs
for which
Distinguishing number and distinguishing index of some operations on graphs
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. We examine the effects
on and when is modified by operations on vertex and edge of
. Let be a connected graph of order . We show that , where denotes the graph obtained from by
removal of a vertex and all edges incident to and these inequalities
are true for the distinguishing index. Also we prove that
and , where denotes the graph obtained from
by simply removing the edge . Finally we consider the vertex contraction
and the edge contraction of and prove that the edge contraction decrease
the distinguishing number (index) of by at most one and increase by at most
().Comment: 11 pages, 4 figure
On the second largest distance eigenvalue of a graph
Let be a simple connected graph of order and be the distance
matrix of Suppose that
are the
distance spectrum of . A graph is said to be determined by its
-spectrum if with respect to the distance matrix , any graph with the
same spectrum as is isomorphic to . In this paper, we consider spectral
characterization on the second largest distance eigenvalue
of graphs, and prove that the graphs with
are determined by
their -spectra
On the lower bounds of Davenport constant
Let with
be a finite abelian group. The Davenport constant is the
smallest integer such that every sequence over of length
has a non-empty zero-sum subsequence. It is a starting point of zero-sum theory
but only has a trivial lower bound , which equals over -groups. We investigate the
non-dispersive sequences over group , thereby revealing the growth of
over non--groups
with . We give a general lower bound of over
non--groups and show that, let be abelian groups with and
rank , fix a non-prime-power, then for each there exists an
such that if , then
Recurrence relations and splitting formulas for the domination polynomial
The domination polynomial D(G,x) of a graph G is the generating function of
its dominating sets. We prove that D(G,x) satisfies a wide range of reduction
formulas. We show linear recurrence relations for D(G,x) for arbitrary graphs
and for various special cases. We give splitting formulas for D(G,x) based on
articulation vertices, and more generally, on splitting sets of vertices
Jones type basic construction on field algebras of -spin models
Let be a finite group. Starting from the field algebra of
-spin models, one can construct the crossed product -algebra
such that it coincides with the -basic
construction for the field algebra and the -invariant
subalgebra of , where is the quantum double of . Under
the natural -module action on ,the
iterated crossed product -algebra can be obtained, which is
-isomorphic to the -basic construction for and the field algebra . Furthermore, one can show that the
iterated crossed product -algebra is a new field algebra and give the
concrete structure with the order and disorder operators.Comment: 14 page
Introduction to Domination Polynomial of a Graph
We introduce a domination polynomial of a graph G. The domination polynomial
of a graph G of order n is the polynomial D(G, x) =\sum_{i=1}^n d(G, i)x^i,
where d(G, i) is the number of dominating sets of G of size i. We obtain some
properties of D(G, x) and its coefficients. Also we compute this polynomial for
some specific graphs.Comment: 10 pages. Accepted
http://www.combinatorialmath.ca/ArsCombinatoria/index.htm
New skew Laplacian energy of a simple digraph
For a simple digraph of order with vertex set , let and denote the out-degree and in-degree of a vertex
in , respectively. Let and
. In this paper we introduce
to be a new kind of skew Laplacian
matrix of , where and is the
skew-adjacency matrix of , and from which we define the skew Laplacian
energy of as the sum of the norms of all the eigenvalues of
. Some lower and upper bounds of the new skew Laplacian
energy are derived and the digraphs attaining these bounds are also determined.Comment: 13 page
Irreducibility and components rigid in moduli of the Hilbert scheme of smooth curves
Denote by the Hilbert scheme of smooth curves, that is
the union of components whose general point corresponds to a smooth irreducible
and non-degenerate curve of degree and genus in . A
component of is rigid in moduli if its image under the
natural map is a one
point set. In this note, we provide a proof of the fact that
has no components rigid in moduli for and ,
from which it follows that the only smooth projective curves embedded in
whose only deformations are given by projective transformations
are the twisted cubic curves. In case , we also prove the
non-existence of a component of rigid in moduli in a
certain restricted range of , and . In the course of the proofs, we
establish the irreducibility of beyond the range which
has been known before.Comment: v2: added (ii) in Cor. 3.6; v3: minor typos correcte
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