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). Let α0(G) be the smallest α ...for which Aα(G) is positive semidefinite. It is known that α0(G)≤1/2. The main results of this paper are:(1)if G is d-regular thenα0=−λmin(A(G))d−λmin(A(G)), where λmin(A(G)) is the smallest eigenvalue of A(G);(2)G contains a bipartite component if and only if α0(G)=1/2;(3)if G is r-colorable, then α0(G)≥1/r.
The α-Hermitian adjacency matrix Hα of a mixed graph X has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of ...the complex number α. This enables us to define an incidence matrix of mixed graphs. Consequently, we define a generalization of line graphs as well as a generalization of the signless Laplacian adjacency matrix of graphs. We then study the spectral properties of the gamma-signless Laplacian adjacency matrix of a mixed graph. Lastly, we characterize when the signless Laplacian adjacency matrix of a mixed graph is singular and give lower and upper bounds of number of arcs and digons in terms of largest and lowest eigenvalue of the signless Laplacian adjacency matrix.
In this paper we introduce a new graph matrix, named the anti-adjacency matrix or eccentricity matrix, which is constructed from the distance matrix of a graph by keeping for each row and each column ...only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping for each row and each column only the distances equal to 1. We show that the eccentricity matrix of trees is irreducible, and we investigate the relations between the eigenvalues of the adjacency and eccentricity matrices. Finally, we give some applications of this new matrix in terms of molecular descriptors, and we conclude by proposing some further research problems.
Recent progress on human action recognition, fueled by the Graph Convolutional Network (GCN), has been substantial. However, two main problems are caused by the design strategy of graph convolution ...kernels: first, the partitioning strategy of neighbor set for graph vertices relies on the gravity center designed manually, which is limited in generalizability to diverse skeletons in action recognition; second, the existing GCN-based methods can only capture local physical dependencies among joints and result in missing implicit joint correlations due to over-smoothing. In this work, we present (1) a novel attention adjacency matrix (AAM) to design graph convolution kernels and (2) a dimension-attention block to improve the robustness of the model. Specifically, the proposed AAM is designed by a novel partitioning strategy for the neighbor set, through which an adjacency matrix is decomposed into several parametric matrices. Simultaneously, attention mechanism is introduced in the process to generate an attention matrix. Combining the matrix and the parametric matrices into an AAM through ResNet, we further exhibit the AAM based graph convolution network (AAM-GCN). The proposed dimension-attention block strengthens the important information in each dimension of skeleton data by extending the idea of channel-attention. Extensive experiments on two large-scale datasets, NTU-RGB+D and Kinetics, demonstrate that AAM-GCN achieves better performance than the state-of-the-art works.
•In the abstract, the correct definition of Aex(G) is added.•The definitions for the matrix A and for the matrix SDD are added.•The adjacency matrix A(G) was defined.•On page 2, after ”For recent ...research along these lines, we added: see 5,20,33 and “Recall that the extended adjacency matrix is just one among a large number of degree-based graph matrices; for details see A.”•On page 10, together with Refs. 17, 21, 19 we quoted also B:r•On page 10, together with Ref. 27 we quoted also Ref A.•Two following refs are added:A K. C. Das, I. Gutman, I. Milovanovi’c, E. Milovanovi’c, B. Furtula, Degree-based energies of graphs, Linear Algebra Appl. 554 (2018) 185–204.B I. Gutman, H. Ramane, Research on graph energies in 2019, MATCH Commun. Math. Comput. Chem. 84 (2020) 277–292.
The extended adjacency matrix of graph G,Aex is a symmetric real matrix that if i≠j and uiuj∈E(G), then the ijth entry is dui2+duj2/2duiduj, and zero otherwise, where du indicates the degree of vertex u. In the present paper, several investigations of the extended adjacency matrix are undertaken and then some spectral properties of Aex are given. Moreover, we present some lower and upper bounds on extended adjacency spectral radii of graphs. Besides, we also study the behavior of the extended adjacency energy of a graph G.
A weighted graph Gω consists of a simple graph G with a weight ω, which is a mapping, ω: E(G)→Z∖{0}. A signed graph is a graph whose edges are labelled with −1 or 1. In this paper, we characterize ...graphs which have a sign such that their signed adjacency matrix has full rank, and graphs which have a weight such that their weighted adjacency matrix does not have full rank. We show that for any arbitrary simple graph G, there is a sign σ so that Gσ has full rank if and only if G has a {1,2}-factor. We also show that for a graph G, there is a weight ω so that Gω does not have full rank if and only if G has at least two {1,2}-factors.
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.
A graph is claw-free if it does not contain a star of order 4 as an induced subgraph. In this paper, we characterize all claw-free connected graphs with all but four eigenvalues equal to 0 or −1.
Based on the Hermitian adjacency matrices of second kind introduced by Mohar 1 and weighted adjacency matrices introduced in 2, we define a kind of index weighted Hermitian adjacency matrices of ...mixed graphs. In this paper we characterize the structure of mixed graphs which are cospectral to their underlying graphs, then we determine a upper bound on the spectral radius of mixed graphs with maximum degree Δ , and characterize the corresponding extremal graphs.
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