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1
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Order unicyclic graphs according to spectral radius of unoriented laplacian matrix

100%
Fan Y.-Z., Wu S.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
|
nr 3
487-499
EN
The spectral radius of a graph is defined by that of its unoriented Laplacian matrix. In this paper, we determine the unicyclic graphs respectively with the third and the fourth largest spectral radius among all unicyclic graphs of given order.
2
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The Laplacian spectrum of some digraphs obtained from the wheel

100%
Su L., Li H.-H., Zheng L.-R.
Discussiones Mathematicae Graph Theory
|
2012
|
tom 32
|
nr 2
255-261
EN
The problem of distinguishing, in terms of graph topology, digraphs with real and partially non-real Laplacian spectra is important for applications. Motivated by the question posed in [R. Agaev, P. Chebotarev, Which digraphs with rings structure are essentially cyclic?, Adv. in Appl. Math. 45 (2010), 232-251], in this paper we completely list the Laplacian eigenvalues of some digraphs obtained from the wheel digraph by deleting some arcs.
3
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An upper bound on the Laplacian spectral radius of the signed graphs

88%
Li H.-H., Li J.-S.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
|
nr 2
345-359
EN
In this paper, we established a connection between the Laplacian eigenvalues of a signed graph and those of a mixed graph, gave a new upper bound for the largest Laplacian eigenvalue of a signed graph and characterized the extremal graph whose largest Laplacian eigenvalue achieved the upper bound. In addition, an example showed that the upper bound is the best in known upper bounds for some cases.
4
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Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix

75%
Kurata H., Bapat R. B.
Special Matrices
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2016
|
tom 4
|
nr 1
270-282
EN
By a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice versa. This correspondence extends a similar correspondence between Euclidean distance matrices and positive semidefinite matrices with zero row and column sums.We show that if B has rank r, then the corresponding D must have rank r, r + 1 or r + 2. We give a complete characterization of the three cases.We obtain formulas for the Moore-Penrose inverse D+ in terms of B+, extending formulas obtained in Kurata and Bapat (Linear Algebra and Its Applications, 2015). If D is the distance matrix of a weighted tree with the sum of the weights being zero, then B+ turns out to be the Laplacian of the tree, and the formula for D+ extends a well-known formula due to Graham and Lovász for the inverse of the distance matrix of a tree.
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