306 results on '"Adjacency matrix"'
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2. On claw-free graphs with all but four eigenvalues equal to 0 or [formula omitted].
- Author
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Sun, Shaowei and Chen, Mengsi
- Subjects
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EIGENVALUES , *GRAPH connectivity , *CLAWS - Abstract
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. [ABSTRACT FROM AUTHOR]
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- 2024
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3. Reciprocal eigenvalue properties using the zeta and Möbius functions.
- Author
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Kadu, Ganesh S., Sonawane, Gahininath, and Borse, Y.M.
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MOBIUS function , *FUNCTION algebras , *EIGENVALUES , *ZETA functions , *LINEAR operators , *VECTOR spaces , *BOOLEAN algebra - Abstract
In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra B n of order 2 n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ (P) of complement-closed and convex subposets P of B n. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V (P) on P. We then write down the adjacency operator A of the graph Γ (P) as the composition of two linear operators on V (P) , namely, the operator that multiplies elements of V (P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P , then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ (P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property. [ABSTRACT FROM AUTHOR]
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- 2024
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4. The unique spectral extremal graph for intersecting cliques or intersecting odd cycles.
- Author
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Miao, Lu, Liu, Ruifang, and Zhang, Jingru
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COMPLETE graphs - Abstract
The (k , r) -fan, denoted by F k , r , is the graph consisting of k copies of the complete graph K r which intersect in a single vertex. Desai et al. [7] proved that E X s p (n , F k , r) ⊆ E X (n , F k , r) for sufficiently large n , where E X s p (n , F k , r) and E X (n , F k , r) are the sets of n -vertex F k , r -free graphs with maximum spectral radius and maximum size, respectively. In this paper, the set E X s p (n , F k , r) is uniquely determined for n large enough. Let H s , t 1 , ... , t k be the graph consisting of s triangles and k odd cycles of lengths t 1 , ... , t k ≥ 5 intersecting in exactly one common vertex, denoted by H s , k for short. Li and Peng [12] showed that E X s p (n , H s , k) ⊆ E X (n , H s , k) for n large enough. In this paper, the set E X s p (n , H s , k) is uniquely characterized for sufficiently large n. [ABSTRACT FROM AUTHOR]
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- 2024
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5. Strong star complements in graphs.
- Author
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Anđelić, Milica, Rowlinson, Peter, and Stanić, Zoran
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REGULAR graphs , *EIGENVALUES - Abstract
Let G be a finite simple graph with λ as an eigenvalue (i.e. an eigenvalue of the adjacency matrix of G), and let H be a star complement for λ in G. Motivated by a controllability condition, we say that H is a strong star complement for λ if G and H have no eigenvalue in common. We explore this concept in the context of line graphs, exceptional graphs, strongly regular graphs and graphs with a prescribed star complement. [ABSTRACT FROM AUTHOR]
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- 2024
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6. On the smallest positive eigenvalue of bipartite graphs with a unique perfect matching.
- Author
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Barik, Sasmita, Behera, Subhasish, and Pati, Sukanta
- Subjects
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BIPARTITE graphs , *EIGENVALUES , *GRAPH connectivity - Abstract
Let G be a simple graph with the adjacency matrix A (G) , and let τ (G) denote the smallest positive eigenvalue of A (G). Let G n be the class of all connected bipartite graphs on n = 2 k vertices with a unique perfect matching. In this article, we characterize the graphs G in G n such that τ (G) does not exceed 1 2. Using the above characterization, we obtain the unique graphs in G n with the maximum and the second maximum τ , respectively. Further, we prove that the largest and the second largest limit points of the smallest positive eigenvalues of bipartite graphs with a unique perfect matching are 1 2 and the reciprocal of α 3 1 2 + α 3 − 1 2 , respectively, where α 3 is the largest root of x 3 − x − 1. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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7. On unimodular graphs with a unique perfect matching.
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Basumatary, Parameswar and Sarma, Kuldeep
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NONNEGATIVE matrices , *GRAPH connectivity , *EIGENVALUES - Abstract
A graph is called unimodular if its adjacency matrix has determinant ± 1. This article provides a necessary and sufficient condition for a simple connected graph with a unique perfect matching to be unimodular. In particular, we give a complete characterization of bicyclic unimodular graphs with a unique perfect matching. Moreover, the possible values of the determinant of the adjacency matrix of unicyclic, bicyclic, and tricyclic graphs with a unique perfect matching are also provided in this article. For non-bipartite unicyclic graphs with a unique perfect matching, we address the problem of when the inverse of the corresponding adjacency matrix is diagonally similar to a non-negative matrix. A pseudo-unimodular graph is a singular graph whose product of non-zero eigenvalues of the corresponding adjacency matrix is ± 1. We supply a necessary and sufficient condition for a singular graph to be pseudo-unimodular. [ABSTRACT FROM AUTHOR]
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- 2024
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8. Spectral extrema of [formula omitted]-free graphs.
- Author
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Zhai, Yanni and Yuan, Xiying
- Abstract
For a set of graphs F , a graph is said to be F -free if it does not contain any graph in F as a subgraph. Let Ex s p (n , F) denote the graphs with the maximum spectral radius among all F -free graphs of order n. A linear forest is a graph whose connected components are paths. Denote by L s the family of all linear forests with s edges. In this paper the graphs in Ex s p (n , { K k + 1 , L s }) will be completely characterized when n is appropriately large. [ABSTRACT FROM AUTHOR]
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- 2024
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9. Upper bounds of spectral radius of symmetric matrices and graphs.
- Author
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Jin, Ya-Lei, Zhang, Jie, and Zhang, Xiao-Dong
- Subjects
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SYMMETRIC matrices , *MATHEMATICAL bounds , *ABSOLUTE value , *EIGENVALUES - Abstract
The spectral radius ρ (A) is the maximum absolute value of the eigenvalues of a matrix A. In this paper, we establish some relationship between the spectral radius of a symmetric matrix and its principal submatrices, i.e., if A is partitioned as a 2 × 2 block matrix A = ( 0 A 12 A 21 A 22 ) , then ρ (A) 2 ≤ ρ 2 2 + θ ⁎ , where θ ⁎ is the largest real root of the equation μ 2 = (x − ν) 2 (ρ 2 2 + x) and ρ 2 = ρ (A 22) , μ = ρ (A 12 A 22 A 21) , ν = ρ (A 12 A 21). Furthermore, the results are used to obtain several upper bounds of the spectral radius of graphs, which strengthen or improve some known results. [ABSTRACT FROM AUTHOR]
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- 2024
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10. An adjacency matrix perspective of talented monoids and Leavitt path algebras.
- Author
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Bock, Wolfgang and Sebandal, Alfilgen N.
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ALGEBRA , *ACYCLIC model , *MONOIDS , *GENERATORS of groups , *MATRICES (Mathematics) , *APERIODICITY - Abstract
In this article we establish relationships between Leavitt path algebras, talented monoids and the adjacency matrices of the underlying graphs. We show that indeed the adjacency matrix generates in some sense the group action on the generators of the talented monoid. With the help of this, we deduce a form of the aperiodicity index of a graph via the talented monoid. We classify hereditary and saturated subsets via the adjacency matrix. This then translates to a correspondence between the composition series of the talented monoid and the so-called matrix composition series of the adjacency matrix. In addition, we discuss the number of cycles in a graph. In particular, we give an equivalent characterization of acyclic graphs via the adjacency matrix, the talented monoid and the Leavitt path algebra. Finally, we compute the number of linearly independent paths of certain length in the Leavitt path algebra via adjacency matrices. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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11. Estimating distance between an eigenvalue of a signed graph and the spectrum of an induced subgraph.
- Author
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Stanić, Zoran
- Subjects
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EIGENVALUES , *EIGENVECTORS , *REGULAR graphs - Abstract
The distance between an eigenvalue λ of a signed graph G ̇ and the spectrum of a signed graph H ̇ is defined as min { | λ − μ | : μ is an eigenvalue of H ̇ }. In this paper, we investigate this distance when H ̇ is a largest induced subgraph of G ̇ that does not have λ as an eigenvalue. We estimate the distance in terms of eigenvectors and structural parameters related to vertex degrees. For example, we show that | λ | | λ − μ | ≤ δ G ̇ ∖ H ̇ max { d H ̇ (i) d G ̇ − V (H ̇) (j) : i ∈ V (G ̇) ∖ V (H ̇) , j ∈ V (H ̇) , i ∼ j } , where δ G ̇ ∖ H ̇ is the minimum vertex degree in V (G ̇) ∖ V (H ̇). If H ̇ is obtained by deleting a single vertex i , this bound reduces to | λ | | λ − μ | ≤ d (i). We also consider the case in which λ is a simple eigenvalue and H ̇ is not necessarily a vertex-deleted subgraph, and the case when λ is the largest eigenvalue of an ordinary (unsigned) graph. Our results for signed graphs apply to ordinary graphs. They can be interesting in the context of eigenvalue distribution, eigenvalue location or spectral distances of (signed) graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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12. Adjacency matrices over a finite prime field and their direct sum decompositions.
- Author
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Higashitani, Akihiro and Sugishita, Yuya
- Subjects
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CONGRUENCES & residues , *MATRIX decomposition , *UNDIRECTED graphs - Abstract
In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field F p. We apply symmetric (row and column) elementary transformations to the adjacency matrix over F p in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over F p for any odd prime p. Our key tool is quadratic residues of F p. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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13. Fractional revival between twin vertices.
- Author
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Monterde, Hermie
- Subjects
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LAPLACIAN matrices , *WEIGHTED graphs - Abstract
In this paper, we provide a characterization of fractional revival between twin vertices in a weighted graph with respect to its adjacency, Laplacian and signless Laplacian matrices. As an application, we characterize fractional revival between apexes of double cones. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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14. On the divisibility of H-shape trees and their spectral determination.
- Author
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Chen, Zhen, Wang, Jianfeng, Brunetti, Maurizio, and Belardo, Francesco
- Subjects
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TREES , *POLYNOMIALS , *DIAMETER - Abstract
A graph G is divisible by a graph H if the characteristic polynomial of G is divisible by that of H. In this paper, a necessary and sufficient condition for recursive graphs to be divisible by a path is used to show that the H-shape graph P 2 , 2 ; n − 4 2 , n − 7 , known to be (for n large enough) the minimizer of the spectral radius among the graphs of order n and diameter n − 5 , is determined by its adjacency spectrum if and only if n ≠ 10 , 13 , 15. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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15. Spectral extrema of graphs with bounded clique number and matching number.
- Author
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Wang, Hongyu, Hou, Xinmin, and Ma, Yue
- Abstract
For a set of graphs F , let ex (n , F) and spex (n , F) denote the maximum number of edges and the maximum spectral radius of an n -vertex F -free graph, respectively. Nikiforov (LAA , 2007) gave the spectral version of the Turán Theorem by showing that spex (n , K k + 1) = λ (T k (n)) , where T k (n) is the k -partite Turán graph on n vertices. In the same year, Feng, Yu and Zhang (LAA) determined the exact value of spex (n , M s + 1) , where M s + 1 is a matching with s + 1 edges. Recently, Alon and Frankl (arXiv:2210.15076) gave the exact value of ex (n , { K k + 1 , M s + 1 }). In this article, we give the spectral version of the result of Alon and Frankl by determining the exact value of spex (n , { K k + 1 , M s + 1 }) when n is large. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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16. A note on the bounds for the spectral radius of graphs.
- Author
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Filipovski, Slobodan and Stevanović, Dragan
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MATHEMATICAL bounds , *UNDIRECTED graphs , *ELECTRONIC information resource searching , *DATABASE searching , *EIGENVALUES , *RAMSEY numbers - Abstract
Let G = (V , E) be a finite undirected graph of order n and of size m. Let Δ and δ be the largest and the smallest degree of G , respectively. The spectral radius of G is the largest eigenvalue of the adjacency matrix of the graph G. In this note we give new bounds on the spectral radius of { C 3 , C 4 } -free graphs in terms of m , n , Δ and δ. Computer search shows that in most of the cases the bounds derived in this note are better than the existing bounds. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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17. The maximum spectral radius of graphs of given size with forbidden subgraph.
- Author
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Fang, Xiaona and You, Lihua
- Abstract
Let G be a graph of size m and ρ (G) be the spectral radius of its adjacency matrix. A graph is said to be F -free if it does not contain a subgraph isomorphic to F. Let θ 1 , 2 , 3 denote the graph obtained from three internally disjoint paths with the same pair of endpoints, where the three paths are of lengths 1, 2, 3, respectively. Recently, Li, Sun and Wei showed that for any θ 1 , 2 , 3 -free graph G of size m ≥ 8 , ρ (G) ≤ 1 + 4 m − 3 2 , with equality if and only if G ≅ S m + 3 2 , 2 , where S m + 3 2 , 2 = K 2 ▿ K m − 1 2 ‾. However, this bound is not attainable when m is even. In this paper, we characterize the graphs with the maximum spectral radius over K 2 , r + 1 -free non-star graphs with m ≥ (4 r + 2) 2 + 1 , the graphs with the maximum spectral radius over θ 1 , 2 , 3 -free graphs with m ≥ 22 when m is even, and the graphs with the second largest spectral radius over θ 1 , 2 , 3 -free graphs with m ≥ 22 when m is odd. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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18. Graphs with second largest eigenvalue less than 1/2.
- Author
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Wu, Xiaoxia, Qian, Jianguo, and Peng, Haigen
- Subjects
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EIGENVALUES , *REAL numbers , *POINT set theory , *GRAPH connectivity - Abstract
We characterize the simple connected graphs with the second largest eigenvalue less than 1/2, which consists of 13 classes of specific graphs. These 13 classes hint that c 2 ∈ [ 1 / 2 , 2 + 5 ] , where c 2 is the minimum real number c for which every real number greater than c is a limit point in the set of the second largest eigenvalues of the simple connected graphs. We leave it as a problem. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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19. On graphs with eigenvectors in {−1,0,1} and the max k-cut problem.
- Author
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Alencar, Jorge, de Lima, Leonardo, and Nikiforov, Vladimir
- Subjects
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EIGENVECTORS , *MATHEMATICAL bounds , *LAPLACIAN matrices , *EIGENVALUES , *MATRICES (Mathematics) - Abstract
In this paper, we characterize all graphs with eigenvectors of the signless Laplacian and adjacency matrices with components equal to { − 1 , 0 , 1 }. We extend the graph parameter max k -cut to square matrices and prove a general sharp upper bound, which implies upper bounds on the max k -cut of a graph using the smallest signless Laplacian eigenvalue, the smallest adjacency eigenvalue, and the largest Laplacian eigenvalue of the graph. In addition, we construct infinite families of extremal graphs for the obtained upper bounds. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
20. Upper bounds on the smallest positive eigenvalue of trees with at most one zero eigenvalue.
- Author
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Rani, Sonu
- Subjects
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EIGENVALUES , *TREES , *SPANNING trees , *MATRICES (Mathematics) - Abstract
In this article, we consider the trees for which the adjacency matrix has nullity at most one. We study the smallest positive (adjacency) eigenvalue of these trees. For such trees lower bounds on the smallest positive eigenvalue have already been obtained in literature but the problem of finding an upper bound still remains unsolved. In this article, we find upper bounds on the smallest positive eigenvalue of these trees and obtain corresponding unique maximal trees. Additionally, we study the smallest positive eigenvalue of trees obtained by attaching a pendant at some vertex of a path graph and also identify the trees for the extremal cases. As applications of our earlier results, we show that, for a path of odd order, the smallest positive eigenvalue is maximum, if we add the pendant almost in the middle. As another application, we show that for a path of any order, the smallest positive eigenvalue is minimum, if we add the pendant vertex at the end. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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21. A bound for the p-domination number of a graph in terms of its eigenvalue multiplicities.
- Author
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Abiad, A., Akbari, S., Fakharan, M.H., and Mehdizadeh, A.
- Subjects
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EIGENVALUES , *REGULAR graphs , *LINEAR algebra , *MULTIPLICITY (Mathematics) , *SET theory , *GRAPH connectivity - Abstract
Let G be a connected graph of order n with domination number γ (G). Wang, Yan, Fang, Geng and Tian [Linear Algebra Appl. 607 (2020), 307-318] showed that for any Laplacian eigenvalue λ of G with multiplicity m G (λ) , it holds that γ (G) ≤ n − m G (λ). Using techniques from the theory of star sets, in this work we prove that the same bound holds when λ is an arbitrary adjacency eigenvalue of a non-regular graph, and we characterize the cases of equality. Moreover, we show a result that gives a relationship between start sets and the p -domination number, and we apply it to extend the aforementioned spectral bound to the p -domination number using the adjacency and Laplacian eigenvalue multiplicities. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
22. On the Laplacian spectrum of k-uniform hypergraphs.
- Author
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Saha, S.S., Sharma, K., and Panda, S.K.
- Subjects
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HYPERGRAPHS , *LAPLACIAN matrices , *EIGENVALUES - Abstract
In this article, we consider the generalization of the Laplacian matrix for hypergraphs to obtain several results related to spectral properties of hypergraphs. This generalization of Laplacian matrix was defined in 2021 by Anirban Banerjee. We first supply a necessary and sufficient condition on the Laplacian spectral radius of hypergraphs such that the complement of that hypergraph is connected. We give bounds for the Laplacian spectral radius of k -uniform hypergraphs in terms of some invariants of hypergraph, such as the maximum degree, minimum degree, first Zagreb index, and the chromatic number. As an application of these results, some upper bounds of the Nordhaus-Gaddum type are obtained for the sum of Laplacian spectral radius of a k -uniform hypegraphs and its complement, and product of Laplacian spectral radius of a k -uniform hypegraphs and its complement. It is known that the second largest Laplacian eigenvalue for graphs is greater or equal to the second highest degree of graphs. We show by an example that this result is not true for 3-uniform hypergraphs in general. Finally, we supply a class of 3-uniform hypergraphs for which the second largest Laplacian eigenvalue is greater or equal to the second highest degree. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
23. On the spectral radius of the generalized adjacency matrix of a digraph.
- Author
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Baghipur, Maryam, Ganie, Hilal A., Ghorbani, Modjtaba, and Andrade, Enide
- Subjects
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MATRICES (Mathematics) , *MATHEMATICAL bounds - Abstract
Let D be a strongly connected digraph and α ∈ [ 0 , 1 ]. In Liu et al. (2019) [13] the matrix A α (D) = α D e g (D) + (1 − α) A (D) , where A (D) and D e g (D) are the adjacency matrix and the diagonal matrix of the out-degrees of D , respectively, was defined. In this paper it is established some sharp bounds on the A α (D) -spectral radius in terms of some parameters such as the out-degrees, the maximum out-degree, the second maximum out-degree, the number of vertices, the number of arcs, the average 2-outdegrees of the vertices of D and the parameter α of A α (D). The extremal digraphs attaining these bounds are characterized. It is shown that the bounds obtained improve, in some cases, some of recently given bounds presented in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
24. Turán problem for [formula omitted]-free signed graphs.
- Author
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Chen, Fan and Yuan, Xiying
- Abstract
Let G ˙ be an unbalanced signed graph. In this paper, we establish that e (G ˙) ⩽ 1 2 (n 2 − n) − (n − 3) and ρ (G ˙) ⩽ n − 2 if G ˙ does not contain unbalanced K 4 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 K 4 − in unbalanced signed graph, and we characterize all the extremal signed graphs. • We solve the spectral Turán problem among all K 4 − -free unbalanced signed graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. Exploring non-Euclidean relationships in EEG emotion recognition: A dynamic graph attention network.
- Author
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Fu, Rongrong, Cai, Mengpu, Wang, Shiwei, Wang, Yaodong, and Jia, Chengcheng
- Subjects
EMOTION recognition ,ELECTROENCEPHALOGRAPHY ,DEEP learning ,HEBBIAN memory ,DIFFERENTIAL entropy ,EMOTIONAL state ,DEEP brain stimulation ,EMOTIONS - Abstract
• A novel dynamic graph attention network is proposed for EEG emotion recognition. • The high accuracy can be obtained from the DGAT model in EEG emotion recognition. • The channel connectivity can reflect the synergistic relationships within the brain. Deep learning classification models based on electroencephalogram (EEG) emotion recognition have demonstrated considerable proficiency in the categorization of emotional states. However, these models have limitations in their capability to analyze the active states and cooperative relationships among distinct brain regions. This study proposes a dynamic graph attention network (DGAT) for EEG emotion recognition, which learns the features of each channel and leverages multiple-head self-attention mechanisms to capture non-Euclidean relationships between channels. Then, we use differential entropy features of emotions signals on the SJTU emotion EEG dataset (SEED). The DGAT model achieved improved subject-dependent and cross-subject classification accuracy compared to previous models. Moreover, ablation studies show that the channel weight matrix(CWM) and appropriate hyper-parameters can improve the performance of the DGAT model significantly. Furthermore, by conducting interpretable analysis of the new connections and electrode weights learned by the model, we find that these connection weight relationships reflect a certain degree of coordination within the brain for EEG-based emotion recognition. These findings provide a new method for EEG emotion recognition and highlights the potential for using deep learning models to analyze the active state and synergistic relationships among brain regions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. Singular graphs and the reciprocal eigenvalue property.
- Author
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Barik, Sasmita, Mondal, Debabrota, Pati, Sukanta, and Sarma, Kuldeep
- Subjects
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EIGENVALUES , *WEIGHTED graphs , *GRAPH connectivity , *REGULAR graphs , *POLYNOMIALS - Abstract
Let G be a simple connected graph and A (G) be its adjacency matrix. The terms singularity, eigenvalues, and characteristic polynomial of G mean those of A (G). A nonsingular graph G is said to have the reciprocal eigenvalue property if the reciprocal of each eigenvalue of G is also an eigenvalue. A graph G (possibly singular) is said to have the weak reciprocal eigenvalue property if the reciprocal of each nonzero eigenvalue of it is also an eigenvalue. In Barik et al. (2022) [3] , the authors proved that there is no nontrivial tree with the weak reciprocal eigenvalue property and posed the following question: "Does there exist a nontrivial graph with the weak reciprocal eigenvalue property?" Suppose that G is singular and the characteristic polynomial of G is x n − k (x k + a 1 x k − 1 + ⋯ + a k). Assume that A (G) has rank k , so that a k ≠ 0. Can we ever have | a k | = 1 ? The answer turns out to be negative. As an application, we settle the question posed in Barik et al. (2022) [3]. Another similar application is also mentioned. It is natural to wonder, "Does there exist a nontrivial, simple, connected weighted graph with the weak reciprocal eigenvalue property?" We provide a class of such graphs. Furthermore, we extend our results to weighted graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. A sharp upper bound on the spectral radius of C5-free /C6-free graphs with given size.
- Author
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Min, Gao, Lou, Zhenzhen, and Huang, Qiongxiang
- Subjects
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CHARTS, diagrams, etc. , *EDGES (Geometry) , *MOTIVATION (Psychology) - Abstract
Let S n , 2 be the graph obtained by joining each vertex of K 2 to n − 2 isolated vertices, and let S n , 2 − be the graph obtained from S n , 2 by deleting an edge incident to a vertex of degree two. Recently, Zhai, Lin and Shu [20] showed that ρ (G) ≤ 1 + 4 m − 3 2 for any C 5 -free graph of size m ≥ 8 or C 6 -free graph of size m ≥ 22 , with equality if and only if G ≅ S m + 3 2 , 2 (possibly, with some isolated vertices). However, this bound is sharp only for odd m. Motivated by this, we want to obtain a sharp upper bound of ρ (G) for C 5 -free or C 6 -free graphs with m edges. In this paper, we prove that if G is a C 5 -free graph of even size m ≥ 14 or C 6 -free graph of even size m ≥ 74 , and G contains no isolated vertices, then ρ (G) ≤ ρ ˜ (m) , with equality if and only if G ≅ S m + 4 2 , 2 − , where ρ ˜ (m) is the largest root of x 4 − m x 2 − (m − 2) x + (m 2 − 1) = 0. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
28. GGNet: A novel graph structure for power forecasting in renewable power plants considering temporal lead-lag correlations.
- Author
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Zhu, Nanyang, Wang, Ying, Yuan, Kun, Yan, Jiahao, Li, Yaping, and Zhang, Kaifeng
- Subjects
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CONVOLUTIONAL neural networks , *GRAPH neural networks , *WIND power plants , *PHOTOVOLTAIC power systems , *POWER plants , *AIR flow - Abstract
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 RMSE N and MAE N of wind power plants for 1–4 h forecast steps decreased on average by 22.925% and 13.18%, respectively; the RMSE N and MAE N 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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. An attention-based adaptive spatial–temporal graph convolutional network for long-video ergonomic risk assessment.
- Author
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Zhou, Chengju, Zeng, Jiayu, Qiu, Lina, Wang, Shuxi, Liu, Pingzhi, and Pan, Jiahui
- Subjects
- *
RISK assessment , *INDUSTRIAL workers , *GRAPH algorithms , *KITCHEN remodeling - Abstract
Ergonomic risk assessment (ERA) is commonly used to identify and analyze postures that are detrimental to the health of workers in industrial workplaces, which is vital to prevent work-related musculoskeletal disorders (WMSDs). Among the automatic approaches, algorithms based on graph convolutional networks (GCNs) have shown promising results in ERA using skeleton sequence as input. However, previous GCN-based methods still have certain limitations. First, the separated modeling of spatial and temporal information and the manually pre-defined topology of graph may restrict the representation diversity of the networks. Additionally, RNN-based temporal modeling often incurs high computational costs and fails to capture long-range temporal dependencies, thereby reducing flexibility in describing long videos. To overcome these challenges, in this study, we propose an attention-based adaptive spatial–temporal graph convolutional network (AAST-GCN), aiming to achieve effective and efficient action representation for ERA in long video. First, we employ an alternate modeling strategy to effectively capture the spatial–temporal information, and propose an improved adaptive adjacency matrix scheme to learn various coordination and relations of body-joints, thus enhancing the flexibility to model diverse postures. Furthermore, we introduce an efficient multi-scale temporal convolutional network as a replacement for RNN-based algorithms, enabling the network to extract various granularities of temporal features. Moreover, to make the network focuses on more valuable information, we employ a spatial–temporal interaction attention (STIA) module. Finally, the aforementioned modules are aggregated within a multi-task learning framework, with the action segmentation serving as the auxiliary task to further improve the accuracy of ERA. We conducted the ergonomic risk assessment on the UW-IOM and TUM Kitchen datasets using our network. Extensive experiments conducted on the most popular datasets UW-IOM and TUM Kitchen demonstrated that our proposed AAST-GCN outperforms other GCN-based methods. Ablation studies and visualization also prove the effectiveness of the individual sub-modules. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
30. A switching method for constructing cospectral gain graphs.
- Author
-
Abiad, Aida, Belardo, Francesco, and Khramova, Antonina P.
- Subjects
- *
GENERALIZATION - Abstract
A gain graph over a group G , also referred to as G -gain graph, is a graph where an element of a group G , called gain, is assigned to each oriented edge, in such a way that the inverse element is associated with the opposite orientation. Gain graphs can be regarded as a generalization of signed graphs, among others. In this work, we show a new switching method to construct cospectral gain graphs. Some previous methods known for graph cospectrality follow as a corollary of our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. On the spectral characterization of the p-sun and the (p,q)-double sun.
- Author
-
Allem, L. Emilio, da Silveira, Lucas G.M., and Trevisan, Vilmar
- Subjects
- *
GENEALOGY , *EIGENVALUES - Abstract
In 1973 Schwenk [7] proved that almost every tree has a cospectral mate. Inspired by Schwenk's result, in this paper we study the spectrum of two families of trees. The p -sun of order 2 p + 1 is a star K 1 , p with an edge attached to each pendant vertex, which we show to be determined by its spectrum among connected graphs. The (p , q) -double sun of order 2 (p + q + 1) is the union of a p -sun and a q -sun by adding an edge between their central vertices. We determine when the (p , q) -double sun has a cospectral mate and when it is determined by its spectrum among connected graphs. Our method is based on the fact that these trees have few distinct eigenvalues and we are able to take advantage of their nullity to shorten the list of candidates. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
32. Spectra of quaternion unit gain graphs.
- Author
-
Belardo, Francesco, Brunetti, Maurizio, Coble, Nolan J., Reff, Nathan, and Skogman, Howard
- Subjects
- *
QUATERNIONS , *GRAPH theory , *SPECTRAL theory , *LAPLACIAN matrices , *EIGENVALUES - Abstract
A quaternion unit gain graph is a graph where each orientation of an edge is given a quaternion unit, which is the inverse of the quaternion unit assigned to the opposite orientation. In this paper we define the adjacency, Laplacian and incidence matrices for a quaternion unit gain graph and study their properties. These properties generalize several fundamental results from spectral graph theory of ordinary graphs, signed graphs and complex unit gain graphs. Bounds for both the left and right eigenvalues of the adjacency and Laplacian matrix are developed, and the right eigenvalues for the cycle and path graphs are explicitly calculated. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
33. A spectral condition for the existence of a pentagon in non-bipartite graphs.
- Author
-
Guo, Hangtian, Lin, Huiqiu, and Zhao, Yanhua
- Subjects
- *
BIPARTITE graphs - Abstract
A graph G is F -free, if it contains no F as a subgraph. A Brualdi-Solheid-Turán type problem asks what is the maximum spectral radius of an F -free graph of order n ? Let K a , b • K 3 denote the graph obtained by identifying a vertex of K a , b belonging to the part of size b and a vertex of K 3. In this paper, we consider about a Brualdi-Solheid-Turán type problem on non-bipartite graphs. We prove that if G is a non-bipartite graph with order n ≥ 21 and ρ (G) ≥ ρ (K ⌈ n − 2 2 ⌉ , ⌊ n − 2 2 ⌋ • K 3) , then G contains a pentagon unless G ≅ K ⌈ n − 2 2 ⌉ , ⌊ n − 2 2 ⌋ • K 3. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
34. Signed graphs with three eigenvalues: Biregularity and beyond.
- Author
-
Rowlinson, Peter and Stanić, Zoran
- Subjects
- *
EIGENVALUES , *GRAPH connectivity , *REGULAR graphs - Abstract
First we investigate net-biregular signed graphs with spectrum of the form [ ρ , μ m , λ l ] where λ is non-main; such graphs are necessarily biregular with exactly two main eigenvalues. We provide two constructions of signed graphs with three eigenvalues, where the graphs that arise include net-biregular and net-regular signed graphs having spectrum [ ρ , μ , λ l ] , with λ non-main. Secondly we determine all the connected signed graphs with spectrum [ ρ , μ 2 , λ l ] (l ≥ 2) where λ is non-main: these include a new infinite family of signed graphs which are neither net-regular nor net-biregular. Thus, in contrast to the situation for graphs, a signed graph with two main eigenvalues and one non-main eigenvalue is not necessarily net-biregular. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
35. A spherical evolution algorithm with two-stage search for global optimization and real-world problems.
- Author
-
Wang, Yirui, Cai, Zonghui, Guo, Lijun, Li, Guoqing, Yu, Yang, and Gao, Shangce
- Subjects
- *
GLOBAL optimization , *SEARCH algorithms , *TIME complexity , *GRAPH theory , *COMPUTATIONAL complexity - Abstract
This paper proposes a spherical evolution algorithm with two-stage search. Spherical search and hypercube search are combined to achieve individuals' evolution. A self-adaptive Gaussian scale factor and a variable scale factor are designed to adaptively control individuals' spherical and hypercube search area according to their search situations. Two search stages frequently switch with four search cases to enhance the balance between exploration and exploitation processes. A directed adjacency matrix is devised to analyze the relationship among individuals from the perspective of graph theory. Experiments compare the proposed algorithm with five algorithms with distinctive search patterns on twenty nine CEC2017 benchmark functions. The diversity analysis and graph theory analysis show the good population diversity and effective information spreading of the proposed algorithm. Twenty two real-world problems evaluate the practicality and optimization ability of the proposed algorithm. Finally, the computational time complexity demonstrates that the proposed algorithm is more efficient than the original spherical evolution algorithm. • A spherical evolution algorithm with two-stage search is proposed. • Spherical search and hypercube search are combined to adaptively guide individuals' evolution. • A directed adjacency matrix is firstly introduced to analyze the relationship among individuals. • Functions and real-world problems verify the superior search performance of the proposed algorithm. • The computational efficiency of the proposed algorithm is enhanced. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. Proof of a conjecture on extremal spectral radii of blow-up graphs.
- Author
-
Lou, Zhenzhen and Zhai, Mingqing
- Subjects
- *
LOGICAL prediction , *EVIDENCE , *INDEPENDENT sets - Abstract
A blow-up of a graph G , is obtained from G by replacing each vertex v i of G with an independent set V i of n i vertices and joining each vertex in V i with each vertex in V j provided v i v j ∈ E (G). Let M n (G) be the set of all the blow-ups of G such that each n i ≥ 1 and ∑ i = 1 | G | n i = n. Monsalve and Rada (2021) [9] proposed three conjectures on the extremal spectral radii of M n (P k) and M n (C k). In this paper, we confirm the first two conjectures and construct an infinite class of graphs to illustrate the third conjecture is not true. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
37. On graphs with adjacency and signless Laplacian matrices eigenvectors entries in {−1,+1}.
- Author
-
Alencar, Jorge and de Lima, Leonardo
- Subjects
- *
EIGENVECTORS , *LAPLACIAN matrices - Abstract
In 1986, Herbert Wilf asked what kind of graphs have an eigenvector with entries formed only by ±1. In this paper, we answer this question for the adjacency, Laplacian, and signless Laplacian matrices of a graph. To this end, we generalize the concept of an exact graph to the adjacency and signless Laplacian matrices. Infinite families of exact graphs for all those matrices are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
38. Godsil-McKay switching for mixed and gain graphs over the circle group.
- Author
-
Belardo, Francesco, Brunetti, Maurizio, Cavaleri, Matteo, and Donno, Alfredo
- Subjects
- *
CIRCLE - Abstract
In this paper we describe two methods, both inspired from Godsil-McKay switching on simple graphs, to build cospectral gain graphs whose gain group consists of the complex numbers of modulus 1 (the circle group). The results obtained here can be also applied to the Hermitian matrix of mixed graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
39. Regional traffic flow combination prediction model considering virtual space of the road network.
- Author
-
Hou, Yue, Zhang, Di, Li, Da, and Deng, Zhiyuan
- Subjects
- *
TRAFFIC flow , *SMOOTHING (Numerical analysis) , *PREDICTION models , *PARTICLE swarm optimization , *TRAFFIC congestion , *MATRIX decomposition , *TECHNOLOGICAL forecasting - Abstract
Accurate traffic flow forecasting is an important technical measure to alleviate traffic congestion. Since traffic flow has spatial and temporal characteristics, thus the adequate extraction of its spatio-temporal features is an important prerequisite to promote the forecast accuracy of the model. However, a majority of existing traffic flow prediction models cannot sufficiently consider the neighborhood spatio-temporal relationship for the real road network in the modeling process, which makes it difficult to improve the model prediction accuracy. For this reason, this paper takes improved feature enhancement graph convolution (FEGC), gated recurrent unit (GRU), and improved lightweight particle swarm optimization (ILPSO) algorithm as components, respectively, to construct a combinatorial traffic flow prediction model FEGC-GRU-ILPSO (FGI), aiming to achieve accurate forecast for regional traffic flow through fully learning spatio-temporal correlation characteristics. Firstly, considering that most traffic flow modeling methods are ineffective in characterizing the hidden association information within nodes, we propose the method of constructing the virtual space adjacency matrix based on the improved gray relational analysis (IGRA) algorithm, which achieves the effective characterization of road network neighborhood relationship by fusing it with the original adjacency matrix. Then, based on the idea of matrix decomposition, the weight adjacency matrix is further introduced to realize the dynamic capture of time-varying correlation of node graph structure in realistic road networks. Secondly, to address the performance degradation problem caused by feature assimilation in multi-layer graph convolution, an improved feature enhancement graph convolution component is proposed to alleviate the multi-layer graph convolution over-smoothing by enhancing salient features. Finally, considering the convex optimization problem caused by the way the hyperparameters of the model are determined through subjective experience, we propose the ILPSO algorithm to improve the overall prediction performance in an adaptive optimizing method. In this paper, real-world data acquired by the Caltrans Performance Measurement System (PeMS) is used as the object of study. The experimental results demonstrate that the FGI model has better prediction performance than the current mainstream baseline models. • An adaptive weighted adjacency matrix is proposed to learn road network features. • A feature enhancement graph convolution is proposed to reinforce important features. • An improved particle swarm algorithm is proposed to optimize the model parameters. • The feature capture process of feature enhancement graph convolution is visualized. • The proposed model has superior prediction accuracy than the baseline models. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. Linear ternary codes of strongly regular signed graphs.
- Author
-
Stanić, Zoran
- Subjects
- *
LINEAR codes , *REGULAR graphs , *MATRICES (Mathematics) , *VECTOR spaces , *FINITE fields - Abstract
We consider linear ternary codes generated by A + β I , where A is the adjacency matrix of a strongly regular signed graph, I is the identity matrix and β ∈ F 3. Our results include theoretical examinations on dimension and distance, and the interplay between the codes. We also compute many structural parameters of codes for some infinite families of strongly regular signed graphs and establish more than 60 particular codes of comparatively small dimension. It occurs that some known linear ternary codes are covered by this approach. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Gain-line graphs via G-phases and group representations.
- Author
-
Cavaleri, Matteo, D'Angeli, Daniele, and Donno, Alfredo
- Subjects
- *
GROUP algebras , *MATRICES (Mathematics) , *LAPLACIAN matrices , *REPRESENTATIONS of graphs , *FOURIER transforms - Abstract
Let G be an arbitrary group. We define a gain-line graph for a gain graph (Γ , ψ) through the choice of an incidence G -phase matrix inducing ψ. We prove that the switching equivalence class of the gain function on the line graph L (Γ) does not change if one chooses a different G -phase inducing ψ or a different representative of the switching equivalence class of ψ. In this way, we generalize to any group some results proven by N. Reff in the abelian case. The investigation of the orbits of some natural actions of G on the set H Γ of G -phases of Γ allows us to characterize gain functions on Γ, gain functions on L (Γ) , their switching equivalence classes and their balance property. The use of group algebra valued matrices plays a fundamental role and, together with the matrix Fourier transform, allows us to represent a gain graph with Hermitian matrices and to perform spectral computations. Our spectral results also provide some necessary conditions for a gain graph to be a gain-line graph. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
42. Bounds for the energy of a complex unit gain graph.
- Author
-
Samanta, Aniruddha and Kannan, M. Rajesh
- Subjects
- *
ODD numbers , *EIGENVALUES - Abstract
A T -gain graph, Φ = (G , φ) , is a graph in which the function φ assigns a unit complex number to each orientation of an edge of G , and its inverse is assigned to the opposite orientation. The associated adjacency matrix A (Φ) is defined canonically. The energy E (Φ) of a T -gain graph Φ is the sum of the absolute values of all eigenvalues of A (Φ). We study the notion of energy of a vertex of a T -gain graph, and establish bounds for it. For any T -gain graph Φ, we prove that 2 τ (G) − 2 c (G) ≤ E (Φ) ≤ 2 τ (G) Δ (G) , where τ (G) , c (G) and Δ (G) are the vertex cover number, the number of odd cycles and the largest vertex degree of G , respectively. Furthermore, using the properties of vertex energy, we characterize the class of T -gain graphs for which E (Φ) = 2 τ (G) − 2 c (G) holds. Also, we characterize the T -gain graphs for which E (Φ) = 2 τ (G) Δ (G) holds. This characterization solves a general version of an open problem. In addition, we establish bounds for the energy in terms of the spectral radius of the associated adjacency matrix. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
43. A conjecture on the eigenvalues of threshold graphs.
- Author
-
Tura, Fernando C.
- Subjects
- *
REGULAR graphs , *EIGENVALUES , *LOGICAL prediction , *GRAPH connectivity - Abstract
Let A n be the connected anti-regular graph of order n. It was conjectured that among all threshold graphs on n vertices, A n has the smallest positive eigenvalue and the largest eigenvalue less than −1. Recently, in [1] was given partial results for this conjecture and identified the critical cases where a more refined method is needed. In this paper, we deal with these cases and confirm that conjecture holds. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
44. Classes of nonbipartite graphs with reciprocal eigenvalue property.
- Author
-
Barik, Sasmita and Pati, Sukanta
- Subjects
- *
EIGENVALUES , *GRAPH connectivity , *BIPARTITE graphs , *RECIPROCALS (Mathematics) - Abstract
Let G be a simple connected graph and A (G) be the adjacency matrix of G. The graph G is said to have the reciprocal eigenvalue property (R) if A (G) is nonsingular and 1 λ is an eigenvalue of A (G) whenever λ is an eigenvalue of A (G). Further, if λ and 1 λ have the same multiplicity, for each eigenvalue λ , then G is said to have the strong reciprocal eigenvalue property (SR). Till date, all the classes of bipartite graphs that are found to have property (R), are found to have property (SR) and it is not known whether these two properties are equivalent even for the bipartite graphs with a unique perfect matching. Among nonbipartite graphs, there is only one known graph class for which these two properties are not equivalent. In this article, we construct some more classes of nonbipartite graphs with property (R) but not (SR). [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
45. Spectra of eccentricity matrices of graphs.
- Author
-
Mahato, Iswar, Gurusamy, R., Kannan, M. Rajesh, and Arockiaraj, S.
- Subjects
- *
MATRICES (Mathematics) , *GRAPH connectivity - Abstract
The eccentricity matrix of a connected graph G is obtained from the distance matrix of G by retaining the largest distances in each row and each column, and setting the remaining entries as 0. In this article, a conjecture about the least eigenvalue of eccentricity matrices of trees, presented in the article (Wang et al., 2018), is solved affirmatively. In addition to this, the spectra and the inertia of eccentricity matrices of various classes of graphs are investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
46. Intrinsic dimension estimation based on local adjacency information.
- Author
-
Qiu, Haiquan, Yang, Youlong, and Li, Benchong
- Subjects
- *
EXPECTED returns , *NEIGHBORHOODS , *ELECTRONIC data processing - Abstract
The intrinsic dimension (ID) of a data set is crucial for data processing, especially for high-dimensional data sets. In order to obtain an accurate ID estimate, two neighborhoods of sample points with a radius ratio of k are considered. The ratio of the number of sample points contained in the two neighborhoods is denoted as q ∊ . When the data set is located on a d -dimensional submanifold in R D , the expected value of q ∊ is k d . Based on this consideration, we redefine the adjacency matrix by using the local adjacency information of sample points and propose a new ID estimation method known as ID(k). The ID(k) algorithm contains only one parameter, the scaling ratio k , and we outline the criterion through which the user can select an appropriate k value. We demonstrate the convergence of the new method both theoretically and experimentally. Experimental results from artificial and real data sets show that the estimates obtained by this new ID(k) method are closer to the true intrinsic dimension than those derived using similar methods. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
47. Integral cographs.
- Author
-
Allem, Luiz Emilio and Tura, Fernando
- Subjects
- *
EIGENVALUES , *INTEGERS , *MATRICES (Mathematics) - Abstract
A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper, we show that a cograph that has a balanced cotree T G (a 1 , ... , a r − 1 , 0 | 0 , ... , 0 , a r) is integral. We present closed formulas for its eigenvalues. As application, we use the integral cographs to estimate the eigenvalues of any cograph. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
48. On hyper-Hamiltonicity in graphs.
- Author
-
Del-Vecchio, Renata R., Vinagre, Cybele T.M., and Pereira, Guilherme B.
- Subjects
- *
HAMILTONIAN graph theory , *LAPLACIAN matrices , *EIGENVALUES - Abstract
A Hamiltonian graph G is said to be hyper-Hamiltonian when G − v is Hamiltonian for any vertex v of G. In this paper, sufficient conditions for a graph to be hyper-Hamiltonian based on different parameters, such as degree and number of edges, are presented. Furthermore, other sufficient conditions for hyper-Hamiltonicity are obtained from spectral parameters, such as the spectral radii of the adjacency matrix, the signless Laplacian matrix and the distance matrix. Results on hyper-Hamiltonicity of a threshold graph through the eigenvalues of its Laplacian matrix are also presented. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
49. On split graphs with four distinct eigenvalues.
- Author
-
Goldberg, Felix, Kirkland, Steve, Varghese, Anu, and Vijayakumar, Ambat
- Subjects
- *
DIAMETER - Abstract
It is a well-known fact that a graph of diameter d has at least d + 1 eigenvalues. A graph is d -extremal, if it has diameter d and exactly d + 1 eigenvalues. A graph is split if its vertex set can be partitioned into a clique and a stable set. Such graphs have diameter at most 3. We obtain a complete classification of the connected bidegreed 3-extremal split graphs using the association of split graphs with combinatorial designs. We also construct certain families of non-bidegreed 3-extremal split graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
50. The spectral Turán problem about graphs with no 6-cycle.
- Author
-
Zhai, Mingqing, Wang, Bing, and Fang, Longfei
- Subjects
- *
EXTREMAL problems (Mathematics) , *RADIUS (Geometry) - Abstract
A spectral version of extremal graph theory problem is as follows: what is the maximum spectral radius of an H -free graph of order n ? Nikiforov posed a conjecture on spectral extremal value of cycles. In the past decade, much attention has been paid to this conjecture and other questions on spectral extremal graphs. In order to approach to Nikiforov's conjecture, we provide a new conjecture on spectral extremal value involving fans and wheels. Moreover, we show it is true for k = 2. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
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