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Minimal trees and monophonic convexity

Jose Cáceres, Ortrud R. Oellermann, M. L. Puertas (2012)

Discussiones Mathematicae Graph Theory

Let V be a finite set and 𝓜 a collection of subsets of V. Then 𝓜 is an alignment of V if and only if 𝓜 is closed under taking intersections and contains both V and the empty set. If 𝓜 is an alignment of V, then the elements of 𝓜 are called convex sets and the pair (V,𝓜 ) is called an alignment or a convexity. If S ⊆ V, then the convex hull of S is the smallest convex set that contains S. Suppose X ∈ ℳ. Then x ∈ X is an extreme point for X if X∖{x} ∈ ℳ. A convex geometry on a finite set is...

Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix

Hiroshi Kurata, Ravindra B. Bapat (2016)

Special Matrices

By a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice...

Multi-faithful spanning trees of infinite graphs

Norbert Polat (2001)

Czechoslovak Mathematical Journal

For an end τ and a tree T of a graph G we denote respectively by m ( τ ) and m T ( τ ) the maximum numbers of pairwise disjoint rays of G and T belonging to τ , and we define t m ( τ ) : = min { m T ( τ ) T is a spanning tree of G } . In this paper we give partial answers—affirmative and negative ones—to the general problem of determining if, for a function f mapping every end τ of G to a cardinal f ( τ ) such that t m ( τ ) ≤ f ( τ ) ≤ m ( τ ) , there exists a spanning tree T of G such that m T ( τ ) = f ( τ ) for every end τ of G .

n ( n - 2 ) .

Duchet, Pierre (1988)

Séminaire Lotharingien de Combinatoire [electronic only]

Near-minimal spanning trees : a scaling exponent in probability models

David J. Aldous, Charles Bordenave, Marc Lelarge (2008)

Annales de l'I.H.P. Probabilités et statistiques

We study the relation between the minimal spanning tree (MST) on many random points and the “near-minimal” tree which is optimal subject to the constraint that a proportion δ of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio of lengths should scale as 1+Θ(δ2). We prove this scaling result in the model of the lattice with random edge-lengths and in the euclidean model.

New bounds on the Laplacian spectral ratio of connected graphs

Zhen Lin, Min Cai, Jiajia Wang (2024)

Czechoslovak Mathematical Journal

Let G be a simple connected undirected graph. The Laplacian spectral ratio of G is defined as the quotient between the largest and second smallest Laplacian eigenvalues of G , which is an important parameter in graph theory and networks. We obtain some bounds of the Laplacian spectral ratio in terms of the number of the spanning trees and the sum of powers of the Laplacian eigenvalues. In addition, we study the extremal Laplacian spectral ratio among trees with n vertices, which improves some known...

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