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Bounds On The Disjunctive Total Domination Number Of A Tree

Michael A. Henning, Viroshan Naicker (2016)

Discussiones Mathematicae Graph Theory

Let G be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, γt(G). A set S of vertices in G is a disjunctive total dominating set of G if every vertex is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it. The disjunctive total domination number, [...] γtd(G) γ t d ( G ) , is the minimum cardinality of such a set. We observe that [...] γtd(G)≤γt(G)...

Bounds on the global offensive k-alliance number in graphs

Mustapha Chellali, Teresa W. Haynes, Bert Randerath, Lutz Volkmann (2009)

Discussiones Mathematicae Graph Theory

Let G = (V(G),E(G)) be a graph, and let k ≥ 1 be an integer. A set S ⊆ V(G) is called a global offensive k-alliance if |N(v)∩S| ≥ |N(v)-S|+k for every v ∈ V(G)-S, where N(v) is the neighborhood of v. The global offensive k-alliance number γ ₒ k ( G ) is the minimum cardinality of a global offensive k-alliance in G. We present different bounds on γ ₒ k ( G ) in terms of order, maximum degree, independence number, chromatic number and minimum degree.

Bounds on the Signed 2-Independence Number in Graphs

Lutz Volkmann (2013)

Discussiones Mathematicae Graph Theory

Let G be a finite and simple graph with vertex set V (G), and let f V (G) → {−1, 1} be a two-valued function. If ∑x∈N|v| f(x) ≤ 1 for each v ∈ V (G), where N[v] is the closed neighborhood of v, then f is a signed 2-independence function on G. The weight of a signed 2-independence function f is w(f) =∑v∈V (G) f(v). The maximum of weights w(f), taken over all signed 2-independence functions f on G, is the signed 2-independence number α2s(G) of G. In this work, we mainly present upper bounds on α2s(G),...

Bounds on the subdominant eigenvalue involving group inverses with applications to graphs

Stephen J. Kirkland, Neumann, Michael, Bryan L. Shader (1998)

Czechoslovak Mathematical Journal

Let A be an n × n symmetric, irreducible, and nonnegative matrix whose eigenvalues are λ 1 > λ 2 ≥ ... ≥ λ n . In this paper we derive several lower and upper bounds, in particular on λ 2 and λ n , but also, indirectly, on μ = max 2 ≤ i ≤ n | λ i | . The bounds are in terms of the diagonal entries of the group generalized inverse, Q # , of the singular and irreducible M-matrix Q = λ 1 I - A . Our starting point is a spectral resolution for Q # . We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected...

Branching random walks on binary search trees: convergence of the occupation measure

Eric Fekete (2010)

ESAIM: Probability and Statistics

We consider branching random walks with binary search trees as underlying trees. We show that the occupation measure of the branching random walk, up to some scaling factors, converges weakly to a deterministic measure. The limit depends on the stable law whose domain of attraction contains the law of the increments. The existence of such stable law is our fundamental hypothesis. As a consequence, using a one-to-one correspondence between binary trees and plane trees, we give a description of the...

Broken Circuits in Matroids-Dohmen’s Inductive Proof

Wojciech Kordecki, Anna Łyczkowska-Hanćkowiak (2013)

Discussiones Mathematicae Graph Theory

Dohmen [4] gives a simple inductive proof of Whitney’s famous broken circuits theorem. We generalise his inductive proof to the case of matroids

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