Journal ArticleDOI
Some results on resistance distances and resistance matrices
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In this paper, an application of resistance distances to the bipartiteness of graphs is given, and an interlacing inequality for eigenvalues of the resistance matrix and the Laplacian matrix is given.Abstract:
In this paper, we obtain formulas for resistance distances and Kirchhoff index of subdivision graphs. An application of resistance distances to the bipartiteness of graphs is given. We also give an interlacing inequality for eigenvalues of the resistance matrix and the Laplacian matrix.read more
Citations
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Book ChapterDOI
Graphs and Matrices
TL;DR: In this paper, the adjacency matrix, a matrix of O's and l's, is used to store a graph or digraph in a computer, and certain matrix operations are seen to correspond to digraph concepts.
Journal ArticleDOI
Minimizing Kirchhoff index among graphs with a given vertex bipartiteness
Jia-Bao Liu,Xiang-Feng Pan +1 more
TL;DR: This paper characterize the graph having the minimum Kf(G) values among graphs with a fixed number n of vertices and fixed vertex bipartiteness, 1 ź v b ź n - 3 .
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The normalized Laplacian, degree-Kirchhoff index and the spanning tree numbers of generalized phenylenes
Zhongxun Zhu,Jia-Bao Liu +1 more
TL;DR: In this paper, an explicit closed-form formula for degree-Kirchhoff index and the number of spanning trees of generalized phenylenes are obtained based on the normalized Laplacian spectrum.
Journal ArticleDOI
Resistance distance and Kirchhoff index of R -vertex join and R -edge join of two graphs
TL;DR: The resistance distances and the Kirchhoff index of G 1 { v } G 2 and G 1{ e} G 2 respectively are formulated.
Journal ArticleDOI
Resistance distance-based graph invariants of subdivisions and triangulations of graphs
Yujun Yang,Douglas J. Klein +1 more
TL;DR: In this article, the authors study three resistance distance-based graph invariants: the Kirchhoff index, and two modifications, namely, the multiplicative degree-Kirchhoff indices and the additive degree-kirchhoffs index.
References
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Book
Generalized inverses: theory and applications
Adi Ben-Israel,T. N. E. Greville +1 more
TL;DR: In this paper, the Moore of the Moore-Penrose Inverse is described as a generalized inverse of a linear operator between Hilbert spaces, and a spectral theory for rectangular matrices is proposed.
Journal ArticleDOI
Which graphs are determined by their spectrum
Edwin van Dam,Willem H. Haemers +1 more
TL;DR: For almost all graphs the answer to the question in the title is still unknown as mentioned in this paper, and the cases for which the answer is known are surveyed in the survey of cases where the Laplacian matrix is known.
Posted Content
Which graphs are determined by their spectrum
Edwin van Dam,Willem H. Haemers +1 more
TL;DR: For almost all graphs the answer to the question in the title is still unknown as mentioned in this paper, and the cases for which the answer is known are surveyed in the survey of cases where the Laplacian matrix is known.
Book
Graphs and Matrices
TL;DR: In this article, the authors present a matrix game based on graph games, where the objective is to find the positive definite completion problem in a graph. But the game is not suitable for children.
Journal ArticleDOI
Resistance distance and the normalized Laplacian spectrum
Haiyan Chen,Fuji Zhang +1 more
TL;DR: Not only is it shown the resistance distance can be naturally expressed in terms of the normalized Laplacian eigenvalues and eigenvectors of G, but also a new index which is closely related to the spectrum of the Normalized LaPLacian is introduced.
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