Removable Edges of a Spanning Tree in 3-Connected 3-Regular Graphs

Abstract Let $${{\mathcal G}_{n,r,s}}$$ denote a uniformly random r-regular s-uniform hypergraph on the vertex set {1, 2, … , n}. We establish a threshold result for the existence of a spanning tree in $${{\mathcal G}_{n,r,s}}$$ , restricting to n satisfying the necessary divisibility conditions. Specifically, we show that when s ≥ 5, there is a positive constant ρ(s) such that for any r ≥ 2, the probability that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree tends to 1 if r > ρ(s), and otherwise this probability tends to zero. The threshold value ρ(s) grows exponentially with s. As $${{\mathcal G}_{n,r,s}}$$ is connected with probability that tends to 1, this implies that when r ≤ ρ(s), most r-regular s-uniform hypergraphs are connected but have no spanning tree. When s = 3, 4 we prove that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree with probability that tends to 1, for any r ≥ 2. Our proof also provides the asymptotic distribution of the number of spanning trees in $${{\mathcal G}_{n,r,s}}$$ for all fixed integers r, s ≥ 2. Previously, this asymptotic distribution was only known in the trivial case of 2-regular graphs, or for cubic graphs.

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Complexity of the maximum leaf spanning tree problem on planar and regular graphs

Theoretical Computer Science ◽

10.1016/j.tcs.2016.02.011 ◽

2016 ◽

Vol 626 ◽

pp. 134-143 ◽

Cited By ~ 5

Author(s):

Alexander Reich

Keyword(s):

Spanning Tree ◽

Regular Graphs

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On Normalized Laplacians, Degree-Kirchhoff Index and Spanning Tree of Generalized Phenylene

Symmetry ◽

10.3390/sym13081374 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1374

Author(s):

Umar Ali ◽

Hassan Raza ◽

Yasir Ahmed

Keyword(s):

Structural Properties ◽

Spanning Tree ◽

Spanning Trees ◽

Molecular Graph ◽

Decomposition Theorem ◽

Regular Graphs ◽

Laplacian Spectrum ◽

Kirchhoff Index ◽

Polynomial Decomposition ◽

Number Of Spanning Trees

The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and 2n squares, denoted by Ln6,4,4. In this paper, by using the normalized Laplacian polynomial decomposition theorem, we have investigated the normalized Laplacian spectrum of Ln6,4,4 consisting of the eigenvalues of symmetric tri-diagonal matrices LA and LS of order 4n+1. As an application, the significant formula is obtained to calculate the multiplicative degree-Kirchhoff index and the number of spanning trees of generalized phenylene network based on the relationships between the coefficients and roots.

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