On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs
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In this paper, a closed-form formula for the enumeration of spanning trees in a simple connected graph G is presented, and upper and lower bounds for the Laplacian Estrada index of G are established based on the vertex degrees of G.Abstract:
Abstract As a generalization of the Sierpiński-like graphs, the subdivided-line graph Г(G) of a simple connected graph G is defined to be the line graph of the barycentric subdivision of G. In this paper we obtain a closed-form formula for the enumeration of spanning trees in Г(G), employing the theory of electrical networks. We present bounds for the largest and second smallest Laplacian eigenvalues of Г(G) in terms of the maximum degree, the number of edges, and the first Zagreb index of G. In addition, we establish upper and lower bounds for the Laplacian Estrada index of Г(G) based on the vertex degrees of G. These bounds are also connected with the number of spanning trees in Г(G).read more
Citations
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On Sombor Index
TL;DR: In this article, lower and upper bounds on the Sombor index of graphs by using some graph parameters are presented. And several relations on SombOR index with the first and second Zagreb indices of graphs are obtained.
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An iteration method for computing the total number of spanning trees and its applications in graph theory
TL;DR: It is shown that the method described can be designed a program for obtaining easily the exact number of spanning trees of some models, including ladder-graph with zero clustering coefficient, wheel-graph having nonzero clustering coefficients and so on.
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Reformulated Zagreb Indices of Some Derived Graphs
TL;DR: In this article, the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitic graph, total graph, and paraline graph are established.
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Minimal Driver Nodes for Structural Controllability of Large-Scale Dynamical Systems: Node Classification
TL;DR: The problem of minimal control inputs to affect the system states such that the resulting system is structurally controllable is solved for general nonlinear systems, and a P-order solution is proposed.
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On the spectrum of linear dependence graph of a finite dimensional vector space
Sushobhan Maity,A. K. Bhuniya +1 more
TL;DR: The main contribution of this article is to find eigen values of adjacency matrix, Laplacian matrix and distance matrix of this graph, which is Eulerian if and only if q is odd.
References
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Computers and Intractability: A Guide to the Theory of NP-Completeness
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
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Algebraic connectivity of graphs
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Laplacian matrices of graphs: a survey
TL;DR: In this paper, the authors survey some of the many results known for Laplacian matrices, and present a survey of the most important results in the field of graph analysis.