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Keywords:
algebraic connectivity; Fiedler vector
Summary:
Let $G$ be a $k$-connected graph with $k \ge 2$. A hinge is a subset of $k$ vertices whose deletion from $G$ yields a disconnected graph. We consider the algebraic connectivity and Fiedler vectors of such graphs, paying special attention to the signs of the entries in Fiedler vectors corresponding to vertices in a hinge, and to vertices in the connected components at a hinge. The results extend those in Fiedler's papers Algebraic connectivity of graphs (1973), A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory (1975), and Kirkland and Fallat's paper Perron Components and Algebraic Connectivity for Weighted Graphs (1998).
References:
[1] Bapat, R. B., Kirkland, S. J., Pati, S.: The perturbed Laplacian matrix of a graph. Linear Multilinear Algebra 49 (2001), 219-242. DOI 10.1080/03081080108818697 | MR 1888190 | Zbl 0984.05056
[2] Bapat, R. B., Lal, A. K., Pati, S.: On algebraic connectivity of graphs with at most two points of articulation in each block. Linear Multilinear Algebra 60 (2012), 415-432. DOI 10.1080/03081087.2011.603727 | MR 2903493 | Zbl 1244.05139
[3] Bapat, R. B., Pati, S.: Algebraic connectivity and the characteristic set of a graph. Linear Multilinear Algebra 45 (1998), 247-273. DOI 10.1080/03081089808818590 | MR 1671627 | Zbl 0944.05066
[4] Abreu, N. M. M. de: Old and new results on algebraic connectivity of graphs. Linear Algebra Appl. 423 (2007), 53-73. DOI 10.1016/j.laa.2006.08.017 | MR 2312323 | Zbl 1115.05056
[5] Fiedler, M.: A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czech. Math. J. 25 (1975), 619-633. MR 0387321 | Zbl 0437.15004
[6] Fiedler, M.: Algebraic connectivity of graphs. Czech. Math. J. 23 (1973), 298-305. MR 0318007 | Zbl 0265.05119
[7] Kirkland, S.: Algebraic connectivity. Handbook of Linear Algebra Discrete Mathematics and Its Applications Chapman & Hall/CRC, Boca Raton (2007), 36-1-36-12 L. Hogben et al.
[8] Kirkland, S., Neumann, M., Shader, B. L.: Characteristic vertices of weighted trees via Perron values. Linear Multilinear Algebra 40 (1996), 311-325. DOI 10.1080/03081089608818448 | MR 1384650 | Zbl 0866.05041
[9] Kirkland, S., Fallat, S.: Perron components and algebraic connectivity for weighted graphs. Linear Multilinear Algebra 44 (1998), 131-148. DOI 10.1080/03081089808818554 | MR 1674228 | Zbl 0926.05026
[10] Merris, R.: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197-198 (1994), 143-176. MR 1275613 | Zbl 0802.05053
[11] Nikiforov, V.: The influence of Miroslav Fiedler on spectral graph theory. Linear Algebra Appl. 439 (2013), 818-821. MR 3061737 | Zbl 1282.05145
[12] Pothen, A., Simon, H. D., Liou, K.-P.: Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11 (1990), 430-452. DOI 10.1137/0611030 | MR 1054210 | Zbl 0711.65034
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