On the Aα-spectral radius of a graph Xue, Jie; Lin, Huiqiu; Liu, Shuting ...
Linear algebra and its applications,
08/2018, Letnik:
550
Journal Article
Recenzirano
Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For any real α∈0,1, Nikiforov 3 defined the matrix Aα(G) asAα(G)=αD(G)+(1−α)A(G). The largest ...eigenvalue of Aα(G) is called the Aα-spectral radius of G. In this paper, we give three edge graft transformations on Aα-spectral radius. As applications, we determine the unique graph with maximum Aα-spectral radius among all connected graphs with diameter d, and determine the unique graph with minimum Aα-spectral radius among all connected graphs with given clique number. In addition, some bounds on the Aα-spectral radius are obtained.
The goal of this article is to display a Q-polynomial structure for the Attenuated Space poset Aq(N,M). The poset Aq(N,M) is briefly described as follows. Start with an (N+M)-dimensional vector space ...H over a finite field with q elements. Fix an M-dimensional subspace h of H. The vertex set X of Aq(N,M) consists of the subspaces of H that have zero intersection with h. The partial order on X is the inclusion relation. The Q-polynomial structure involves two matrices A,A⁎∈MatX(C) with the following entries. For y,z∈X the matrix A has (y,z)-entry 1 (if y covers z); qdimy (if z covers y); and 0 (if neither of y,z covers the other). The matrix A⁎ is diagonal, with (y,y)-entry q−dimy for all y∈X. By construction, A⁎ has N+1 eigenspaces. By construction, A acts on these eigenspaces in a (block) tridiagonal fashion. We show that A is diagonalizable, with 2N+1 eigenspaces. We show that A⁎ acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that A is Q-polynomial. We show that A,A⁎ satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra T of MatX(C) generated by A,A⁎. We show that A,A⁎ act on each irreducible T-module as a Leonard pair.
Let G˙ be an unbalanced signed graph. In this paper, we establish that e(G˙)⩽12(n2−n)−(n−3) and ρ(G˙)⩽n−2 if G˙ does not contain unbalanced K4 as a signed subgraph. Moreover, comprehensive ...characterizations of the extremal signed graphs have been obtained.
•The situation that the index of signed graph equals to the spectral radius of underlying graph has been characterized.•We determine the Turán number of K4− in unbalanced signed graph, and we characterize all the extremal signed graphs.•We solve the spectral Turán problem among all K4−-free unbalanced signed graphs.
In this note we show that for each positive integer a⩾2 there exist infinitely many trees whose spectral radius is equal to 2a. Such trees are obtained by replacing the central edge of the double ...star S(a,2a−2) with suitable bidegreed caterpillars.
Many problems in real world, either natural or man-made, can be usefully represented by graphs or networks. Along with a complex topological structure, the weight is a vital factor in characterizing ...some properties of real networks. In this paper, we define a class of the weighted edge corona product networks. The generalized adjacency (resp., Laplacian and signless Laplacian) spectra with two different structures are determined. As applications, the number of spanning trees and Kirchhoff index of the weighted edge corona product networks are computed.
•We first define a class of the weighted edge corona product networks.•Three types of spectra for that networks are given.•The number of spanning trees and Kirchhoff index of that networks are determined.
Let
$ \mathbb {F} $
F
be a field and suppose
$ \mathbf {a} := (a_1, a_2, \dotsc ) $
a
:=
(
a
1
,
a
2
,
...
)
is a sequence of non-zero elements in
$ \mathbb {F} $
F
. For a tournament T on
$ n $
n
..., associate the
$ n \times n $
n
×
n
symmetric matrix
$ M_{T}(\mathbf {a}) $
M
T
(
a
)
(resp. skew-symmetric matrix
$ M_{T, \mathrm {skew}}(\mathbf {a}) $
M
T
,
skew
(
a
)
) with zero diagonal as follows: for i<j, if the edge ij is directed as
$ i \to j $
i
→
j
in T, then set
$ M_{T}(\mathbf {a}) = a_i $
M
T
(
a
)
=
a
i
(resp.
$ M_{T, \mathrm {skew}}(\mathbf {a}) = a_i $
M
T
,
skew
(
a
)
=
a
i
), else set
$ M_{T}(\mathbf {a}) = a_j $
M
T
(
a
)
=
a
j
(resp.
$ M_{T, \mathrm {skew}}(\mathbf {a}) = a_j $
M
T
,
skew
(
a
)
=
a
j
). Let
$ \mathcal {M}_{n}(\mathbf {a}) $
M
n
(
a
)
(resp.
$ \mathcal {M}_{n, \mathrm {skew}}(\mathbf {a}) $
M
n
,
skew
(
a
)
) be the family consisting of all the
$ n \times n $
n
×
n
symmetric matrices
$ M_{T}(\mathbf {a}) $
M
T
(
a
)
(resp. skew-symmetric matrices
$ M_{T, \mathrm {skew}}(\mathbf {a}) $
M
T
,
skew
(
a
)
) as T varies over all tournaments on
$ n $
n
. We show that any matrix in
$ \mathcal {M}_n(\mathbf {a}) $
M
n
(
a
)
or
$ \mathcal {M}_{n, \mathrm {skew}}(\mathbf {a}) $
M
n
,
skew
(
a
)
corresponding to a transitive tournament has a rank at least n−1, and this is best possible. This settles in a strong form a conjecture posed in Balachandran et al. An ensemble of high-rank matrices arising from tournaments; Linear Algebra Appl. 2023;658:310-318. As a corollary, any matrix in these families has rank at least
$ \lfloor \log _2(n) \rfloor $
⌊
log
2
(
n
)
⌋
.
In this paper, we give a characterization of regular two-graphs of order n in terms of spectral data of two-graphs of order n−1 and n−2. Moreover, we prove that a two-graph of order n is regular if ...and only if all its induced sub-two-graphs with n−2 vertices have the same characteristic polynomial.
It is shown that the nilpotency of submatrices of a certain class of adjacency matrices is equivalent to the Collatz conjecture. Our main result extends the previous work of Alves et al. and ...clarifies a conjecture made by Cardon and Tuckfield.
On the Aα-spectra of trees Nikiforov, Vladimir; Pastén, Germain; Rojo, Oscar ...
Linear algebra and its applications,
05/2017, Letnik:
520
Journal Article
Recenzirano
Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α∈0,1, define the matrix Aα(G) asAα(G)=αD(G)+(1−α)A(G).
This paper gives several ...results about the Aα-matrices of trees. In particular, it is shown that if TΔ is a tree of maximal degree Δ, then the spectral radius of Aα(TΔ) satisfies the tight inequalityρ(Aα(TΔ))<αΔ+2(1−α)Δ−1, which implies previous bounds of Godsil, Lovász, and Stevanović. The proof is deduced from some new results about the Aα-matrices of Bethe trees and generalized Bethe trees.
In addition, several bounds on the spectral radius of Aα of general graphs are proved, implying tight bounds for paths and Bethe trees.
Although eigenvectors belong to the core of linear algebra, relatively few closed-form expressions exist, which we bundle and discuss here. A particular goal is their interpretation for graph-related ...matrices, such as the adjacency matrix of an undirected, possibly weighted graph.
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