Displaying similar documents to “Connectivity, toughness, spanning trees of bounded degree, and the spectrum of regular graphs”

On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs

Yilun Shang (2016)

Open Mathematics

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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....

Note: Sharp Upper and Lower Bounds on the Number of Spanning Trees in Cartesian Product of Graphs

Jernej Azarija (2013)

Discussiones Mathematicae Graph Theory

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Let G1 and G2 be simple graphs and let n1 = |V (G1)|, m1 = |E(G1)|, n2 = |V (G2)| and m2 = |E(G2)|. In this paper we derive sharp upper and lower bounds for the number of spanning trees τ in the Cartesian product G1 □G2 of G1 and G2. We show that: [...] and [...] . We also characterize the graphs for which equality holds. As a by-product we derive a formula for the number of spanning trees in Kn1 □Kn2 which turns out to be [...] .

Some properties of the distance Laplacian eigenvalues of a graph

Mustapha Aouchiche, Pierre Hansen (2014)

Czechoslovak Mathematical Journal

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The distance Laplacian of a connected graph G is defined by ℒ = Diag ( Tr ) - 𝒟 , where 𝒟 is the distance matrix of G , and Diag ( Tr ) is the diagonal matrix whose main entries are the vertex transmissions in G . The spectrum of ℒ is called the distance Laplacian spectrum of G . In the present paper, we investigate some particular distance Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance Laplacian eigenvalues. We establish some properties...

Small integral trees.

Brouwer, A.E. (2008)

The Electronic Journal of Combinatorics [electronic only]

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On the multiplicity of Laplacian eigenvalues of graphs

Ji-Ming Guo, Lin Feng, Jiong-Ming Zhang (2010)

Czechoslovak Mathematical Journal

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In this paper we investigate the effect on the multiplicity of Laplacian eigenvalues of two disjoint connected graphs when adding an edge between them. As an application of the result, the multiplicity of 1 as a Laplacian eigenvalue of trees is also considered.