G
is a simple connected graph with adjacency matrix
A
(
G
) and degree diagonal matrix
D
(
G
). The signless Laplacian matrix of
G
is defined as
Q
(
G
) =
D
(
G
) +
A
(
G
). In 2017, Nikiforov 1 ...defined the matrix
A
α
(
G
) =
α D
(
G
) + (
1
−
α
)
A
(
G
) for
α
∈ 0,1. The
A
α
-spectral radius of
G
is the maximum eigenvalue of
A
α
(
G
). In 2019, Liu
et al.
2 defined the matrix
Θ
k
(
G
) as
Θ
k
(
G
) =
kD
(
G
) +
A
(
G
), for
k
∈ ℝ. In this paper, we present a new type of lower bound for the
A
α
-spectral radius of a graph after vertex deletion. Furthermore, we deduce some corollaries on
Θ
k
(
G
),
A
(
G
),
Q
(
G
) matrices.
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, write Aα(G) for the matrixAα(G)=αD(G)+(1−α)A(G). This paper presents some ...extremal results about the spectral radius ρα(G) of Aα(G) that generalize previous results about ρ0(G) and ρ1/2(G).
In particular, write Bp,q,r be the graph obtained from a complete graph Kp by deleting an edge and attaching paths Pq and Pr to its ends. It is shown that if α∈0,1) and G is a graph of order n and diameter at least k, thenρα(G)≤ρα(Bn−k+2,⌊k/2⌋,⌈k/2⌉), with equality holding if and only if G=Bn−k+2,⌊k/2⌋,⌈k/2⌉. This result generalizes results of Hansen and Stevanović 5, and Liu and Lu 7.
Let G=(V(G),E(G)) be a simple graph with vertex set V(G) and edge set E(G), c(G)=|E(G)|−|V(G)|+θ(G) be the dimension of cycle spaces of G with θ(G) the number of connected components of G, m(G) be ...the matching number of G, and η(G) be the nullity of G. It was shown in Wang and Wong (2014) that the nullity η(G) of G is bounded by an upper bound and a lower bound as |V(G)|−2m(G)−c(G)≤η(G)≤|V(G)|−2m(G)+2c(G). Graphs with nullity attaining the upper bound and the lower bound have been characterized by Song et al. (2015) and Wang (2016), respectively.
In this paper, we prove that there is no graph with nullity η(G)=|V(G)|−2m(G)+2c(G)−1. For fixed c(G), infinitely many connected graphs with nullity |V(G)|−2m(G)+2c(G)−k,(0≤k≤3c(G),k≠1) are also constructed.
The relationship among every pair of vertices in a graph can be represented as a matrix, such as in adjacency matrix and distance matrix. Both adjacency and distance matrices have the same property. ...Adjacency and distance matrices are both symmetric matrix with diagonals entries equals to 0. In this paper, we discuss relationships between adjacency matrix and distance matrix of a graph of diameter two, which is D=2(J-I)-A. From this relationship, we also determine the value of the determinant matrix A+D and the upper bound of determinant of matrix D.
Abstract
Different methods to address the isomorphism in kinematic chains existed earlier and is well reported. In all the methods, link-adjacency matrix (LAM) represents the structural information ...of a given kinematic chain. Having gathered a large collection of co-spectral chains belonging to a large number of chain categories possessing (containing up to 20 links and up to 13 degrees of freedom), the present work aims at addressing the isomorphism by developing and implementing a methodology for isomorphism based on characteristic coefficients of the degree matrix and structural matrix representations of a specified Simple Jointed Kinematic Chain (SJKC) being computed and comparing. While the above two tests proved to be successful in most cases, two instances of cospectral cases were discovered wherein the above polynomial-based tests failed to detect dissimilarity in the chain structures. These cases are reported in the paper.
Power forecast for each renewable power plant (RPP) in the renewable energy clusters is essential. Though existing graph neural networks (GNN)-based models achieve satisfactory prediction performance ...by capturing dependencies among distinct RPPs, the static graph structure employed in these models ignores crucial lead-lag correlations among RPPs, resulting from the time difference of the air flow at spatially dispersed RPPs. To address this problem, this paper proposes a novel dynamic graph structure using multiple temporal granularity groups (TGGs) to characterize the lead-lag correlations among RPPs. A granular-based GNN called GGNet is designed to generate an optimal adjacency matrix for the proposed graph structure. Specifically, a two-dimensional convolutional neural network (2D-CNN) is used to quantify the uncertain lead-lag correlations among RPPs; secondly, a gate mechanism is used to calculate a dynamic adjacency matrix; Finally, a graph attention network (GAT) is used to aggregate the information on RPPs based on the well-learned adjacency matrix. Case studies conducted using real-world datasets, with wind power plants and photovoltaic power plants, show our method outperforms state-of-the-art (SoTA) ones with better performance. Compared with the SoTA models, the RMSEN and MAEN of wind power plants for 1–4 h forecast steps decreased on average by 22.925% and 13.18%, respectively; the RMSEN and MAEN of photovoltaic power plants for 1–4 h forecast steps decreased on average by 48.95% and 18.75%, respectively. The results show that the proposed framework can generate improved performance with accuracy and robustness.
•Dynamic graph structure can clarify lead-lag power correlation of renewable plants.•A novel model can quantity the lead-lag power correlation of renewable plants.•Memory cell can make the adjacency matrix among renewable plants learnable.•A graph attention network can improve power forecast accuracy of renewable plants.
Precise forecasting of wind speed is vital to guarantee safe travel on bridges. Nevertheless, current research primarily concentrates on single-point prediction. When it comes to forecasting spatial ...wind fields, graph neural networks prove to be powerful methods, but many of them have yet to take into account the impact of frequency domain attributes. To address this, a novel model called the graph neural network with time-varying frequency features (GTF) is proposed. Its core layer employs wavelet transforms in matrix form to extract various frequency sub-series, followed by the use of adaptive adjacency matrices to identify their spatial characteristics. Furthermore, aggregating all frequency band information and exploring spatial features further through multi-kernel convolutions. The predictive performance of different models for wind speed forecasting is assessed by utilizing spatial wind field data collected on a large-span bridge. Experimental results demonstrate that spatiotemporal model GTF outperforms the other models in terms of predictive accuracy. The quantity of wind observation points and their level of correlation exert a certain impact on the GTF model, while hyperparameters such as stacking depth and wavelet decomposition parameters have minimal effects. Finally, recommended parameters for the GTF model are provided.
•A graph neural network involving time-varying frequency domain features is proposed.•Atual spatial wind field measurements are utilized for evaluating the proposed model.•The impact of spatial wind field characteristics and model parameters were examined.•The suggested model exhibits commendable precision and stability.
Eigenvectors of random matrices: A survey O'Rourke, Sean; Vu, Van; Wang, Ke
Journal of combinatorial theory. Series A,
November 2016, 2016-11-00, Letnik:
144
Journal Article
Recenzirano
Odprti dostop
Eigenvectors of large matrices (and graphs) play an essential role in combinatorics and theoretical computer science. The goal of this survey is to provide an up-to-date account on properties of ...eigenvectors when the matrix (or graph) is random.
Many algorithms have been proposed for fitting network models with communities, but most of them do not scale well to large networks, and often fail on sparse networks. Here we propose a new fast ...pseudo-likelihood method for fitting the stochastic block model for networks, as well as a variant that allows for an arbitrary degree distribution by conditioning on degrees. We show that the algorithms perform well under a range of settings, including on very sparse networks, and illustrate on the example of a network of political blogs. We also propose spectral clustering with perturbations, a method of independent interest, which works well on sparse networks where regular spectral clustering fails, and use it to provide an initial value for pseudolikelihood. We prove that pseudo-likelihood provides consistent estimates of the communities under a mild condition on the starting value, for the case of a block model with two communities.
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