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An upper bound on the Laplacian spectral radius of the signed graphs

100%
Li H.-H., Li J.-S.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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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.
2
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The Minimum Spectral Radius of Signless Laplacian of Graphs with a Given Clique Number

81%
Su L., Li H.-H., Zhang J.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
|
nr 1
95-102
EN
In this paper we observe that the minimal signless Laplacian spectral radius is obtained uniquely at the kite graph PKn−ω,ω among all connected graphs with n vertices and clique number ω. In addition, we show that the spectral radius μ of PKm,ω (m ≥ 1) satisfies [...] More precisely, for m > 1, μ satisfies the equation [...] where [...] and [...] . At last the spectral radius μ(PK∞,ω) of the infinite graph PK∞,ω is also discussed.
3
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On The Determinant of q-Distance Matrix of a Graph

81%
Li H.-H., Su L., Zhang J.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
|
nr 1
103-111
EN
In this note, we show how the determinant of the q-distance matrix Dq(T) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85-88]. Further, by means of the result, we determine the determinant of the q-distance matrix of the graph obtained from a connected weighted graph G by adding the weighted branches to G, and so generalize in part the results obtained by Bapat et al. [R.B. Bapat, S. Kirkland and M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl. 401 (2005) 193- 209]. In particular, as a consequence, determinantal formulae of q-distance matrices for unicyclic graphs and one class of bicyclic graphs are presented.
4
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The Laplacian spectrum of some digraphs obtained from the wheel

81%
Su L., Li H.-H., Zheng L.-R.
Discussiones Mathematicae Graph Theory
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2012
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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.
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