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Статті в журналах з теми "THEORY OF SIGNED GRAPHS"

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Автор: Grafiati

Опубліковано: 11 вересня 2023

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1

Hou, Yaoping, and Dijian Wang. "Laplacian integral subcubic signed graphs." Electronic Journal of Linear Algebra 37 (February 26, 2021): 163–76. http://dx.doi.org/10.13001/ela.2021.5699.

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Анотація:

A (signed) graph is called Laplacian integral if all eigenvalues of its Laplacian matrix are integers. In this paper, we determine all connected Laplacian integral signed graphs of maximum degree 3; among these signed graphs,there are two classes of Laplacian integral signed graphs, one contains 4 infinite families of signed graphs and another contains 29 individual signed graphs.

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2

Belardo, Francesco, and Maurizio Brunetti. "Connected signed graphs L-cospectral to signed ∞-graphs." Linear and Multilinear Algebra 67, no. 12 (July 9, 2018): 2410–26. http://dx.doi.org/10.1080/03081087.2018.1494122.

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3

Li, Yu, Meng Qu, Jian Tang, and Yi Chang. "Signed Laplacian Graph Neural Networks." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 4 (June 26, 2023): 4444–52. http://dx.doi.org/10.1609/aaai.v37i4.25565.

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Анотація:

This paper studies learning meaningful node representations for signed graphs, where both positive and negative links exist. This problem has been widely studied by meticulously designing expressive signed graph neural networks, as well as capturing the structural information of the signed graph through traditional structure decomposition methods, e.g., spectral graph theory. In this paper, we propose a novel signed graph representation learning framework, called Signed Laplacian Graph Neural Network (SLGNN), which combines the advantages of both. Specifically, based on spectral graph theory and graph signal processing, we first design different low-pass and high-pass graph convolution filters to extract low-frequency and high-frequency information on positive and negative links, respectively, and then combine them into a unified message passing framework. To effectively model signed graphs, we further propose a self-gating mechanism to estimate the impacts of low-frequency and high-frequency information during message passing. We mathematically establish the relationship between the aggregation process in SLGNN and signed Laplacian regularization in signed graphs, and theoretically analyze the expressiveness of SLGNN. Experimental results demonstrate that SLGNN outperforms various competitive baselines and achieves state-of-the-art performance.

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4

Zhang, Xianhang, Hanchen Wang, Jianke Yu, Chen Chen, Xiaoyang Wang, and Wenjie Zhang. "Polarity-based graph neural network for sign prediction in signed bipartite graphs." World Wide Web 25, no. 2 (February 16, 2022): 471–87. http://dx.doi.org/10.1007/s11280-022-01015-4.

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Анотація:

AbstractAs a fundamental data structure, graphs are ubiquitous in various applications. Among all types of graphs, signed bipartite graphs contain complex structures with positive and negative links as well as bipartite settings, on which conventional graph analysis algorithms are no longer applicable. Previous works mainly focus on unipartite signed graphs or unsigned bipartite graphs separately. Several models are proposed for applications on the signed bipartite graphs by utilizing the heuristic structural information. However, these methods have limited capability to fully capture the information hidden in such graphs. In this paper, we propose the first graph neural network on signed bipartite graphs, namely Polarity-based Graph Convolutional Network (PbGCN), for sign prediction task with the help of balance theory. We introduce the novel polarity attribute to signed bipartite graphs, based on which we construct one-mode projection graphs to allow the GNNs to aggregate information between the same type nodes. Extensive experiments on five datasets demonstrate the effectiveness of our proposed techniques.

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5

Tupper, Melissa, and Jacob A. White. "Online list coloring for signed graphs." Algebra and Discrete Mathematics 33, no. 2 (2022): 151–72. http://dx.doi.org/10.12958/adm1806.

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Анотація:

We generalize the notion of online list coloring to signed graphs. We define the online list chromatic number of a signed graph, and prove a generalization of Brooks' Theorem. We also give necessary and sufficient conditions for a signed graph to be degree paintable, or degree choosable. Finally, we classify the 2-list-colorable and 2-list-paintable signed graphs.

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6

DIAO, Y., G. HETYEI, and K. HINSON. "TUTTE POLYNOMIALS OF TENSOR PRODUCTS OF SIGNED GRAPHS AND THEIR APPLICATIONS IN KNOT THEORY." Journal of Knot Theory and Its Ramifications 18, no. 05 (May 2009): 561–89. http://dx.doi.org/10.1142/s0218216509007075.

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Анотація:

It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Our result enables us to compute the Jones polynomials of some large non-alternating knots. The combinatorics used to prove our main result is similar to Tutte's original way of counting "activities" and specializes to a new, perhaps simpler proof of the known formulas for the ordinary Tutte polynomial of the tensor product of unsigned graphs or matroids.

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7

Hameed, Shahul K., T. V. Shijin, P. Soorya, K. A. Germina, and Thomas Zaslavsky. "Signed distance in signed graphs." Linear Algebra and its Applications 608 (January 2021): 236–47. http://dx.doi.org/10.1016/j.laa.2020.08.024.

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8

Acharya, B. D. "Signed intersection graphs." Journal of Discrete Mathematical Sciences and Cryptography 13, no. 6 (December 2010): 553–69. http://dx.doi.org/10.1080/09720529.2010.10698314.

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9

Li, Shu, and Jianfeng Wang. "Yet More Elementary Proof of Matrix-Tree Theorem for Signed Graphs." Algebra Colloquium 30, no. 03 (August 29, 2023): 493–502. http://dx.doi.org/10.1142/s1005386723000408.

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Анотація:

A signed graph [Formula: see text] is a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], together with a function [Formula: see text] assigning a positive or negative sign to each edge. In this paper, we present a more elementary proof for the matrix-tree theorem of signed graphs, which is based on the relations between the incidence matrices and the Laplcians of signed graphs. As an application, we also obtain the results of Monfared and Mallik about the matrix-tree theorem of graphs for signless Laplacians.

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10

Brown, John, Chris Godsil, Devlin Mallory, Abigail Raz, and Christino Tamon. "Perfect state transfer on signed graphs." Quantum Information and Computation 13, no. 5&6 (May 2013): 511–30. http://dx.doi.org/10.26421/qic13.5-6-10.

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Анотація:

We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. First, we show that the signed join of a negative $2$-clique with any positive $(n,3)$-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as $n$ increases. Next, we prove that a signed complete graph has perfect state transfer if its positive subgraph is a regular graph with perfect state transfer and its negative subgraph is periodic. This shows that signing is useful for creating perfect state transfer since no complete graph (except for the $2$-clique) has perfect state transfer. Also, we show that the double-cover of a signed graph has perfect state transfer if the positive subgraph has perfect state transfer and the negative subgraph is periodic.Here, signing is useful for constructing unsigned graphs with perfect state transfer. Finally, we study perfect state transfer on a family of signed graphs called the exterior powers which is derived from a many-fermion quantum walk on graphs.

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11

Mulas, Raffaella, and Zoran Stanić. "Star complements for ±2 in signed graphs." Special Matrices 10, no. 1 (January 1, 2022): 258–66. http://dx.doi.org/10.1515/spma-2022-0161.

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Анотація:

Abstract In this article, we investigate connected signed graphs which have a connected star complement for both − 2 -2 and 2 (i.e. simultaneously for the two eigenvalues), where − 2 -2 (resp. 2) is the least (largest) eigenvalue of the adjacency matrix of a signed graph under consideration. We determine all such star complements and their maximal extensions (again, relative to both eigenvalues). As an application, we provide a new proof of the result which identifies all signed graphs that have no eigenvalues other than − 2 -2 and 2.

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12

Yu, Guihai, Lihua Feng, and Hui Qu. "Signed graphs with small positive index of inertia." Electronic Journal of Linear Algebra 31 (February 5, 2016): 232–43. http://dx.doi.org/10.13001/1081-3810.1976.

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Анотація:

In this paper, the signed graphs with one positive eigenvalue are characterized, and the signed graphs with pendant vertices having exactly two positive eigenvalues are determined. As a consequence, the signed trees, the signed unicyclic graphs and the signed bicyclic graphs having one or two positive eigenvalues are characterized.

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13

Wang, Dijian, and Yaoping Hou. "Integral signed subcubic graphs." Linear Algebra and its Applications 593 (May 2020): 29–44. http://dx.doi.org/10.1016/j.laa.2020.01.037.

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14

Kato, Keiju. "Interior polynomial for signed bipartite graphs and the HOMFLY polynomial." Journal of Knot Theory and Its Ramifications 29, no. 12 (October 2020): 2050077. http://dx.doi.org/10.1142/s0218216520500777.

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Анотація:

The interior polynomial is a Tutte-type invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to the maximal [Formula: see text]-degree part of the HOMFLY polynomial of a naturally associated link. Note that the latter can be any oriented link. This result fits into a program aimed at deriving the HOMFLY polynomial from Floer homology. We also establish some other, more basic properties of the signed interior polynomial. For example, the HOMFLY polynomial of the mirror image of [Formula: see text] is given by [Formula: see text]. This implies a mirroring formula for the signed interior polynomial in the planar case. We prove that the same property holds for any bipartite graph and the same graph with all signs reversed. The proof relies on Ehrhart reciprocity applied to the so-called root polytope. We also establish formulas for the signed interior polynomial inspired by the knot theoretical notions of flyping and mutation. This leads to new identities for the original unsigned interior polynomial.

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15

Guo, Qiao, Yaoping Hou, and Deqiong Li. "The least Laplacian eigenvalue of the unbalanced unicyclic signed graphs with $k$ pendant vertices." Electronic Journal of Linear Algebra 36, no. 36 (June 18, 2020): 390–99. http://dx.doi.org/10.13001/ela.2020.5077.

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Анотація:

Let $\Gamma=(G,\sigma)$ be a signed graph and $L(\Gamma)=D(G)-A(\Gamma)$ be the Laplacian matrix of $\Gamma$, where $D(G)$ is the diagonal matrix of vertex degrees of the underlying graph $G$ and $A(\Gamma)$ is the adjacency matrix of $\Gamma$. It is well-known that the least Laplacian eigenvalue $\lambda_n$ is positive if and only if $\Gamma$ is unbalanced. In this paper, the unique signed graph (up to switching equivalence) which minimizes the least Laplacian eigenvalue among unbalanced connected signed unicyclic graphs with $n$ vertices and $k$ pendant vertices is characterized.

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16

Fan, Yi-Zheng, Wen-Xue Du, and Chun-Long Dong. "The nullity of bicyclic signed graphs." Linear and Multilinear Algebra 62, no. 2 (March 7, 2013): 242–51. http://dx.doi.org/10.1080/03081087.2013.771638.

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17

Deng, Qingying, Xian’an Jin, and Louis H. Kauffman. "Graphical virtual links and a polynomial for signed cyclic graphs." Journal of Knot Theory and Its Ramifications 27, no. 10 (September 2018): 1850054. http://dx.doi.org/10.1142/s0218216518500542.

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Анотація:

For a signed cyclic graph [Formula: see text], we can construct a unique virtual link [Formula: see text] by taking the medial construction and converting 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In this paper, we shall prove that a virtual link [Formula: see text] is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial [Formula: see text] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between [Formula: see text] of a signed cyclic graph [Formula: see text] and the bracket polynomial of one of the virtual link diagrams associated with [Formula: see text]. Finally, we give a spanning subgraph expansion for [Formula: see text].

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18

Simić, Slobodan K., and Zoran Stanić. "Polynomial reconstruction of signed graphs." Linear Algebra and its Applications 501 (July 2016): 390–408. http://dx.doi.org/10.1016/j.laa.2016.03.036.

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19

Stanić, Zoran. "Notes on exceptional signed graphs." Ars Mathematica Contemporanea 18, no. 1 (September 24, 2020): 105–15. http://dx.doi.org/10.26493/1855-3974.1933.2df.

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20

Ghorbani, Ebrahim, Willem H. Haemers, Hamid Reza Maimani, and Leila Parsaei Majd. "On sign-symmetric signed graphs." Ars Mathematica Contemporanea 19, no. 1 (November 10, 2020): 83–93. http://dx.doi.org/10.26493/1855-3974.2161.f55.

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21

Sinha, Deepa, and Ayushi Dhama. "Unitary Addition Cayley Ring Signed Graphs *." Journal of Discrete Mathematical Sciences and Cryptography 18, no. 5 (September 3, 2015): 559–79. http://dx.doi.org/10.1080/09720529.2015.1013680.

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22

Belardo, Francesco, Sebastian M. Cioabă, Jack Koolen, and Jianfeng Wang. "Open problems in the spectral theory of signed graphs." Art of Discrete and Applied Mathematics 1, no. 2 (August 7, 2019): #P2.10. http://dx.doi.org/10.26493/2590-9770.1286.d7b.

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23

Akbari, S., S. Dalvandi, F. Heydari, and M. Maghasedi. "On the eigenvalues of signed complete graphs." Linear and Multilinear Algebra 67, no. 3 (December 26, 2018): 433–41. http://dx.doi.org/10.1080/03081087.2017.1403548.

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24

Sinha, Deepa, Anita Kumari Rao, and Ayushi Dhama. "Spectral analysis of t-path signed graphs." Linear and Multilinear Algebra 67, no. 9 (May 25, 2018): 1879–97. http://dx.doi.org/10.1080/03081087.2018.1472737.

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25

Cameron, P. J., J. J. Seidel, and S. V. Tsaranov. "Signed Graphs, Root Lattices, and Coxeter Groups." Journal of Algebra 164, no. 1 (February 1994): 173–209. http://dx.doi.org/10.1006/jabr.1994.1059.

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26

SimiÄ, Slobodan, and Zoran Stanic. "Polynomial reconstruction of signed graphs whose least eigenvalue is close to -2." Electronic Journal of Linear Algebra 31 (February 5, 2016): 740–53. http://dx.doi.org/10.13001/1081-3810.3245.

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Анотація:

The polynomial reconstruction problem for simple graphs has been considered in the literature for more than forty years and is not yet resolved except for some special classes of graphs. Recently, the same problem has been put forward for signed graphs. Here, the reconstruction of the characteristic polynomial of signed graphs whose vertex-deleted subgraphs have least eigenvalue greater than $-2$ is considered.

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27

Wang, Haichao, Liying Kang, and Erfang Shan. "Signed clique-transversal functions in graphs." International Journal of Computer Mathematics 87, no. 11 (September 2010): 2398–407. http://dx.doi.org/10.1080/00207160902822330.

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28

Mojdeh, D. A., and Babak Samadi. "New and improved results on the signed (total) k-domination number of graphs." Discrete Mathematics, Algorithms and Applications 09, no. 02 (April 2017): 1750024. http://dx.doi.org/10.1142/s1793830917500240.

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Анотація:

In this paper, we study the signed [Formula: see text]-domination and its total version in graphs. By a simple uniform approach we give some new upper and lower bounds on these two parameters of a graph in terms of several different graph parameters. In this way, we can improve and generalize some results in literature. Moreover, we make use of the well-known theorem of Turán [On an extremal problem in graph theory, Math. Fiz. Lapok 48 (1941) 436–452] to bound the signed total [Formula: see text]-domination number, [Formula: see text], of a [Formula: see text]-free graph [Formula: see text] for [Formula: see text].

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29

Lee, Shyi-Long, and Chiuping Li. "Chemical signed graph theory." International Journal of Quantum Chemistry 49, no. 5 (February 15, 1994): 639–48. http://dx.doi.org/10.1002/qua.560490509.

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30

Vijayakumar, G. R. "From finite line graphs to infinite derived signed graphs." Linear Algebra and its Applications 453 (July 2014): 84–98. http://dx.doi.org/10.1016/j.laa.2014.03.047.

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31

Lu, You, Jian Cheng, Rong Luo, and Cun-Quan Zhang. "Shortest circuit covers of signed graphs." Journal of Combinatorial Theory, Series B 134 (January 2019): 164–78. http://dx.doi.org/10.1016/j.jctb.2018.06.001.

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32

Chen, Yu, and Yaoping Hou. "Eigenvalue multiplicity in cubic signed graphs." Linear Algebra and its Applications 630 (December 2021): 95–111. http://dx.doi.org/10.1016/j.laa.2021.08.002.

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33

Belardo, Francesco, and Paweł Petecki. "Spectral characterizations of signed lollipop graphs." Linear Algebra and its Applications 480 (September 2015): 144–67. http://dx.doi.org/10.1016/j.laa.2015.04.022.

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34

Akbari, Saieed, Willem H. Haemers, Hamid Reza Maimani, and Leila Parsaei Majd. "Signed graphs cospectral with the path." Linear Algebra and its Applications 553 (September 2018): 104–16. http://dx.doi.org/10.1016/j.laa.2018.04.021.

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35

Sinha, Deepa, and Ayushi Dhama. "Negation switching invariant 3-Path signed graphs." Journal of Discrete Mathematical Sciences and Cryptography 20, no. 3 (April 3, 2017): 703–16. http://dx.doi.org/10.1080/09720529.2016.1187959.

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36

Brunetti, Maurizio, and Francesco Belardo. "Line graphs of complex unit gain graphs with least eigenvalue -2." Electronic Journal of Linear Algebra 37, no. 37 (February 3, 2021): 14–30. http://dx.doi.org/10.13001/ela.2021.5249.

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Анотація:

Let $\mathbb T$ be the multiplicative group of complex units, and let $\mathcal L (\Phi)$ denote a line graph of a $\mathbb{T}$-gain graph $\Phi$. Similarly to what happens in the context of signed graphs, the real number $\min Spec(A(\mathcal L (\Phi))$, that is, the smallest eigenvalue of the adjacency matrix of $\mathcal L(\Phi)$, is not less than $-2$. The structural conditions on $\Phi$ ensuring that $\min Spec(A(\mathcal L (\Phi))=-2$ are identified. When such conditions are fulfilled, bases of the $-2$-eigenspace are constructed with the aid of the star complement technique.

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37

Lee, Chuan-Min. "Algorithmic Aspects of Some Variations of Clique Transversal and Clique Independent Sets on Graphs." Algorithms 14, no. 1 (January 13, 2021): 22. http://dx.doi.org/10.3390/a14010022.

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Анотація:

This paper studies the maximum-clique independence problem and some variations of the clique transversal problem such as the {k}-clique, maximum-clique, minus clique, signed clique, and k-fold clique transversal problems from algorithmic aspects for k-trees, suns, planar graphs, doubly chordal graphs, clique perfect graphs, total graphs, split graphs, line graphs, and dually chordal graphs. We give equations to compute the {k}-clique, minus clique, signed clique, and k-fold clique transversal numbers for suns, and show that the {k}-clique transversal problem is polynomial-time solvable for graphs whose clique transversal numbers equal their clique independence numbers. We also show the relationship between the signed and generalization clique problems and present NP-completeness results for the considered problems on k-trees with unbounded k, planar graphs, doubly chordal graphs, total graphs, split graphs, line graphs, and dually chordal graphs.

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38

Pardo, Eduardo G., Mauricio Soto, and Christopher Thraves. "Embedding signed graphs in the line." Journal of Combinatorial Optimization 29, no. 2 (March 28, 2013): 451–71. http://dx.doi.org/10.1007/s10878-013-9604-1.

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39

Zhou, Qiannan, and Yong Lu. "Relation between the row left rank of a quaternion unit gain graph and the rank of its underlying graph." Electronic Journal of Linear Algebra 39 (April 20, 2023): 181–98. http://dx.doi.org/10.13001/ela.2023.7681.

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Анотація:

Let $\Phi=(G,U(\mathbb{Q}),\varphi)$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph), where $G$ is the underlying graph of $\Phi$, $U(\mathbb{Q})=\{z\in \mathbb{Q}: |z|=1\}$ is the circle group, and $\varphi:\overrightarrow{E}\rightarrow U(\mathbb{Q})$ is the gain function such that $\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$. Let $A(\Phi)$ be the adjacency matrix of $\Phi$ and $r(\Phi)$ be the row left rank of $\Phi$. In this paper, we prove that $-2c(G)\leq r(\Phi)-r(G)\leq 2c(G)$, where $r(G)$ and $c(G)$ are the rank and the dimension of cycle space of $G$, respectively. All corresponding extremal graphs are characterized. The results will generalize the corresponding results of signed graphs (Lu et al. [20] and Wang [33]), mixed graphs (Chen et al. [7]), and complex unit gain graphs (Lu et al. [21]).

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40

Naserasr, Reza, Lan Anh Pham, and Zhouningxin Wang. "Density of C−4-critical signed graphs." Journal of Combinatorial Theory, Series B 153 (March 2022): 81–104. http://dx.doi.org/10.1016/j.jctb.2021.11.002.

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41

Sun, Gaoxing, Feng Liu, and Kaiyang Lan. "A note on eigenvalues of signed graphs." Linear Algebra and its Applications 652 (November 2022): 125–31. http://dx.doi.org/10.1016/j.laa.2022.07.010.

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42

Belardo, Francesco, and Yue Zhou. "Signed Graphs with extremal least Laplacian eigenvalue." Linear Algebra and its Applications 497 (May 2016): 167–80. http://dx.doi.org/10.1016/j.laa.2016.02.028.

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43

Belardo, Francesco, Irene Sciriha, and Slobodan K. Simić. "On eigenspaces of some compound signed graphs." Linear Algebra and its Applications 509 (November 2016): 19–39. http://dx.doi.org/10.1016/j.laa.2016.07.008.

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44

Stanić, Zoran. "Bounding the largest eigenvalue of signed graphs." Linear Algebra and its Applications 573 (July 2019): 80–89. http://dx.doi.org/10.1016/j.laa.2019.03.011.

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Alikhani, Saeid, Fatemeh Ramezani, and Ebrahim Vatandoost. "On the signed domination number of some Cayley graphs." Communications in Algebra 48, no. 7 (February 6, 2020): 2825–32. http://dx.doi.org/10.1080/00927872.2020.1722830.

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Balakrishnan, Kannan, Manoj Changat, Henry Martyn Mulder, and Ajitha R. Subhamathi. "Consensus strategies for signed profiles on graphs." Ars Mathematica Contemporanea 6, no. 1 (June 15, 2012): 127–45. http://dx.doi.org/10.26493/1855-3974.244.120.

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Zhuo, Kefan, Zhuoxuan Yang, Guan Yan, Kai Yu, and Wenqiang Guo. "An efficient graph clustering algorithm in signed graph based on modularity maximization." International Journal of Modern Physics C 30, no. 11 (November 2019): 1950095. http://dx.doi.org/10.1142/s0129183119500955.

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Анотація:

The unsigned graphs containing positive links only, have been analyzed fruitfully. However, the physical relations behind complex networks are dissimilar. We often encounter the signed networks that have both positive and negative links as well. It is very important to study the characteristics of complex networks and predict individual attitudes by analyzing the attitudes of individuals and their neighbors, which can divide individuals into different clusters or communities. To detect the clusters in signed networks, first, a modularity function for signed networks is proposed on the basis of the combination of positive and negative part. Then, a new graph clustering algorithm for signed graphs has also been proposed based on CNM algorithm, which has high efficiency. Finally, the algorithm has been applied on both artificial and the real networks. The results show that the proposed method has been able to achieve near-perfect solution, which is suitable for multiple types real networks.

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Ahangar, H. Abdollahzadeh, J. Amjadi, S. M. Sheikholeslami, L. Volkmann, and Y. Zhao. "Signed Roman edge domination numbers in graphs." Journal of Combinatorial Optimization 31, no. 1 (May 13, 2014): 333–46. http://dx.doi.org/10.1007/s10878-014-9747-8.

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Ahangar, H. Abdollahzadeh, L. Asgharsharghi, S. M. Sheikholeslami, and L. Volkmann. "Signed mixed Roman domination numbers in graphs." Journal of Combinatorial Optimization 32, no. 1 (April 28, 2015): 299–317. http://dx.doi.org/10.1007/s10878-015-9879-5.

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JIN, XIAN'AN, FENGMING DONG, and ENG GUAN TAY. "DETERMINING THE COMPONENT NUMBER OF LINKS CORRESPONDING TO LATTICES." Journal of Knot Theory and Its Ramifications 18, no. 12 (December 2009): 1711–26. http://dx.doi.org/10.1142/s0218216509007671.

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Анотація:

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.

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