Graph-Theoretic Techniques for the Study of Structures or Networks in EngineeringView this Special Issue
Research Article | Open Access
Algebraic Connectivity and Disjoint Vertex Subsets of Graphs
It is well known that the algebraic connectivity of a graph is the second small eigenvalue of its Laplacian matrix. In this paper, we mainly research the relationships between the algebraic connectivity and the disjoint vertex subsets of graphs, which are presented through some upper bounds on algebraic connectivity.
A graph is often used to model a complex network. The vertex set and the edge set of graph are denoted by and , respectively. A network is represented as an undirected graph consisting of nodes and links, respectively.
Graph theory has provided chemists with a variety of useful tools, such as in the topological structure [1–3]. The Laplacian matrix of a graph is denoted by , and , where is a diagonal matrix whose diagonal entries are its degrees and is the adjacency matrix of . The Laplacian eigenvalues of a graph are the eigenvalues of , denoted by , which are all real and nonnegative. The second smallest Laplacian eigenvalue of a graph is well known as the algebraic connectivity, which was first studied by Fiedler . The algebraic connectivity  of a graph is important for the connectivity of a graph , which can be used to measure the robustness of a graph. It has been emerged as an important parameter in many system problems [7–18]. Especially, the algebraic connectivity also plays an important role in the partitions of a complex network. For the literature on the algebraic connectivity of a graph , the reader is referred to [20, 21]. In this work, the relationships are researched between the algebraic connectivity and disjoint vertex subsets of graphs, which are presented through some upper bounds.
Let be a vector. Let be an incidence matrix of . Then, . For any vector , the inner product of and is defined as . Two upper bounds on the algebraic connectivity are given as follows.
Lemma 1 (see ). For any vector , the Rayleigh inequality is as follows:where , is a constant, and , is the vector for the node .
Lemma 2 (see ). For any vector , we havewhere is the vector for the node .
Let and be two disjoint subsets of , respectively. The distance between two disjoint subsets and of is denoted by . For continence, takes the place of . Let be the distance between the node and , which is the shortest distance of the node to a node of the set . Suppose and . A result on the algebraic connectivity and two partitions of graphs is presented by Alon  and Milman  below.
Lemma 3 (see ). For any two disjoint subsets and of , it holdswhere and are the number of links in the sets and , respectively.
Moreover, the next step consider the case of three disjoint vertex subsets of graphs .
3. Main Result
Let , , and be the subsets of , respectively, where their numbers of nodes are, respectively, , , and . Assume . Let be the distances from the node to subsets of , respectively. Suppose . Now, we construct a function related to node as follows, where the constructed function is referred to the book :
Case 1. If the node belongs to any one subset of , then
Case 2. If the node , then we can see thatBy Case 1 and Case 2, holds. In contrast, if , then for each and , which is a contradiction with . From the definition , for any two adjacent nodes , we haveOur main result is as follows.
Theorem 1. Let be three disjoint subsets of . Let and and be the numbers of links in the sets A and B and C, respectively. Then,
Proof. For subsets , and , by Lemma 2, we haveFrom (2) andwhere , and since the coordinates of the center of gravity of the three regions are the average of the triangle region, then the vectors . The sum of the vectors of the center of gravity of the triangle to the vertices is equal to 0 . The center of gravity is analogous to the mean or average from statistics [6, 31, 32]:By the above inequalities and Lemma 2, it arrives that
Example 1. Figure 1 describes the graphs and , each with nodes, links, and a diameter . For subsets, . For subsets, . Their algebraic connectivity  and their upper bounds on (11) are as follows. For the and aplacian matrixes,For , the algebraic connectivity is 0.6338, and the algebraic connectivity of is 0.5858. For upper bounds on .
For upper bounds on .
A graph with the second smallest Laplacian eigenvalue is thus more robust, in the sense of being better connected.
Proposition 1. Let be three disjoint subsets of . Suppose and . Let , , , and be the number of links in the sets , and , respectively. Then,wherein which is the average of the field.
Proof. For subsets , and , by Lemma 2, we havewhere links of the sets in the node sets , , , and are , , and , respectively. From (2), we obtainBy direct computation, we haveBy the above equalities and Lemma 2, inequality (17) holds.
But, we note that the algebraic connectivity [34, 35], , should not be seen as a strict disconnection or a robustness metric .
Example 2. For example, Figure 2 describes the graphs and , with , , and diameter 6. By direct calculation, for subsets, , and for subsets, . Their algebraic connectivity is 0.4798 and is 0.4817, respectively. Their Laplacian matrices and , for upper bounds on and for upper bounds on .
Theorem 1 and Proposition 1 are two completely different situations. The theorem hypothesis is that be three disjoint subsets of . The proposition supposes that be three disjoint subsets of and and . In other words, the proposition has constraints. Moreover, it is not the same as the four disjoint subsets of .
All data, models, and codes generated or used during the study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the Chunhui project of the Ministry of Education, China (no. Z2017046) and the Qinghai Science and Technology Planning Project (Grant no. 2018-ZJ-718).
- R. García-Domenech, J. Gálvez, and L. Pogliani, “Some new trends in chemical graph theory,” Chemical Reviews, vol. 108, no. 3, pp. 1127–1169, 2008.
- J.-B. de Julián-Ortiz, C. Wang, S. Wang, and B. Wei, “Zagreb indices and multiplicative zagreb indices of eulerian graphs,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 42, no. 1, pp. 67–78, 2019.
- J.-B. Liu, J. Zhao, J. Min, and J. D. Cao, “On the Hosoya index of graphs formed by a fractal graph,” Fractals-Complex Geometry Patterns and Scaling in Nature and Society, vol. 27, no. 8, pp. 105–135, 2019.
- M. Fiedler, “Algebraic connectivity of graphs, Czech,” Journal of Mathematics, vol. 23, pp. 298–305, 1973.
- N. L. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, UK, 2nd edition, 1993.
- J.-B. Liu, J. Zhao, and Z.-Q. Cai, “On the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks,” Physica A: Statistical Mechanics and Its Applications, vol. 540, p. 123073, 2020.
- R. Baeza-Yates and B. Ribeiro-Neto, Modern Information Retrieval, Addison-Wesley, Boston, MA, USA, 1999.
- P. Baldi, P. Frasconi, and P. Smyth, Modeling the Internet and the Web: Probabilistic Methods and Algorithms, John Wiley & Sons, Hoboken, NJ, USA, 2003.
- S. Barnett, Matrices: Methods and Applications, Oxford University Press, Oxford, UK, 1992.
- T. F. Chan, P. Ciarlet, and W. K. Szeto, “On the optimality of the median cut spectral bisection graph partitioning method,” SIAM Journal on Scientific Computing, vol. 18, no. 3, pp. 943–948, 1997.
- A. K. Chandra, P. Raghavan, W. L. Ruzzo, R. Smolensky, and P. Tiwari, “The electrical resistance of a graph captures its commute and cover times,” in Proceedings of the twenty-first annual ACM symposium on Theory of computing—STOC ‘89, pp. 574–586, Seattle WA, USA, May 1989.
- P. Chebotarev and E. Shamis, “The matrix-forest theorem and measuring relations in small social groups,” Automation and Remote Control, vol. 58, no. 9, pp. 1505–1514, 1997.
- P. Chebotarev and E. Shamis, “On a duality between metrics and s-proximities,” Automation and Remote Control, vol. 59, no. 4, pp. 608–612, 1998.
- K.-W. Cheung, K.-C. Tsui, and J. Liu, “Extended latent class models for collaborative recommendation,” IEEE Transactions on Systems, Man, and Cybernetics—Part A: Systems and Humans, vol. 34, no. 1, pp. 143–148, 2004.
- D. Harel and Y. Koren, “On clustering using random walks,” FST TCS 2001: Foundations of Software Technology and Theoretical Computer Science, vol. 22, no. 45, pp. 18–41, 2001.
- J. L. Herlocker, J. A. Konstan, L. G. Terveen, and J. T. Riedl, “Evaluating collaborative filtering recommender systems,” ACM Transactions on Information Systems (TOIS), vol. 22, no. 1, pp. 5–53, 2004.
- N.-D. Ho and P. Van Dooren, “On the pseudo-inverse of the laplacian of a bipartite graph,” Applied Mathematics Letters, vol. 18, no. 8, pp. 917–922, 2005.
- Z. Huang, H. Chen, and D. Zeng, “Applying associative retrieval techniques to alleviate the sparsity problem in collaborative filtering,” ACM Transactions on Information Systems (TOIS), vol. 22, no. 1, pp. 116–142, 2004.
- R. Brooks, “Spectral geometry and the Cheeger constant,” in Expanding Graphs, J. Friedman, Ed., pp. 5–19, American Mathematical Society, Providence, RI, USA, 1993.
- P. Van Mieghem, Graph Spectra for Complex Networks, Delft University of Technology, Delft, Netherlands, 2012.
- N. M. M. de Abreu, “Old and new results on algebraic connectivity of graphs,” Linear Algebra and Its Applications, vol. 423, no. 1, pp. 53–73, 2007.
- N. Alon, “Eigenvalues and expanders,” Combinatorica, vol. 6, no. 2, pp. 83–96, 1986.
- N. Alon and V. D. Milman, “λ1, Isoperimetric inequalities for graphs, and superconcentrators,” Journal of Combinatorial Theory, Series B, vol. 38, no. 1, pp. 73–88, 1985.
- P. K. Chan, M. Schlag, and J. Zien, “Spectral k-way ratio cut partitioning and clustering,” in Proceedings of the Symposium on Research on integrated systems, Warwick, UK, 1993.
- A Court, Nathan College Geometry, An Introduction to the Modern Geometry of the Triangle and the Circle, Barnes & Noble, New York, NY, USA, 2nd edition, 1925.
- F. Chung, Spectral Graph Theory, AMS Publications, Providence, RI, USA, 1997.
- F. R. K. Chung, “Laplacians of graphs and Cheeger inequalities,” in Combinatorics, Paul Erdo is Eighty, pp. 157–172, János Bolyai Mathematical Society, Budapest, Hungary, 1996.
- P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Birkhauser, Basel, Switzerland, 1992.
- J. Cheeger, “A lower bound for the smallest eigenvalue of the Laplacian,” in Problems in Analysis, R. C. Gunnig, Ed., pp. 195–199, Princeton University Press, Princeton, NJ, USA, 1970.
- C. Delorme and S. Poljak, “Laplacian eigenvalues and the maximum cut problem,” Mathematical Programming, vol. 62, no. 1-3, pp. 557–574, 1993.
- J.-B. Liu, J. Zhao, H. He, and Z. Shao, “Valency-based topological descriptors and structural property of the generalized sierpiński networks,” Journal of Statistical Physics, vol. 177, no. 6, pp. 1131–1147, 2019.
- J.-B. Liu, Z.-Yu Shi, Y.-H. Pan, J. Cao, and M. Abdel-Aty, “Udai Al-juboori computing the laplacian spectrum of linear octagonal-quadrilateral networks and its applications,” Polycyclic Aromatic Compounds, vol. 10, pp. 23–43, 2020.
- D. M. Cvetkovic, M. Doob, I. Gutman et al., Recent Results in the Theory of Graph Spectra, Annals of Discrete Mathematics, vol. 36, Elsevier, Amsterdam, Netherlands, 1988.
- C. Godsil and G. Royle, “Algebraic graph theory,” in Graduate Texts in Mathematics, p. 207, Springer-Verlag, New York, NY, USA, 2001.
- S. J. Kirkland, J. J. Molitierno, M. Neumann, and B. L. Shader, “On graphs with equal algebraic and vertex connectivity,” Linear Algebra and Its Applications, vol. 341, no. 1-3, pp. 45–56, 2002.
- G. A. Edgar, D. H. Ullman, and D. B. West, “Problems and solutions,” The American Mathematical Monthly, vol. 125, no. 1, pp. 81–89, 2018.
Copyright © 2020 Yan Sun and Faxu Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.