Displaying similar documents to “Bounds on the subdominant eigenvalue involving group inverses with applications to graphs”

On a bound on algebraic connectivity: the case of equality

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

Czechoslovak Mathematical Journal

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In a recent paper the authors proposed a lower bound on 1 - λ i , where λ i , λ i ≠ 1 , is an eigenvalue of a transition matrix T of an ergodic Markov chain. The bound, which involved the group inverse of I - T , was derived from a more general bound, due to Bauer, Deutsch, and Stoer, on the eigenvalues of a stochastic matrix other than its constant row sum. Here we adapt the bound to give a lower bound on the algebraic connectivity of an undirected graph, but principally consider the case of equality in...

Coalescing Fiedler and core vertices

Didar A. Ali, John Baptist Gauci, Irene Sciriha, Khidir R. Sharaf (2016)

Czechoslovak Mathematical Journal

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The nullity of a graph G is the multiplicity of zero as an eigenvalue in the spectrum of its adjacency matrix. From the interlacing theorem, derived from Cauchy’s inequalities for matrices, a vertex of a graph can be a core vertex if, on deleting the vertex, the nullity decreases, or a Fiedler vertex, otherwise. We adopt a graph theoretical approach to determine conditions required for the identification of a pair of prescribed types of root vertices of two graphs to form a cut-vertex...

Bounds on Laplacian eigenvalues related to total and signed domination of graphs

Wei Shi, Liying Kang, Suichao Wu (2010)

Czechoslovak Mathematical Journal

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A total dominating set in a graph G is a subset X of V ( G ) such that each vertex of V ( G ) is adjacent to at least one vertex of X . The total domination number of G is the minimum cardinality of a total dominating set. A function f : V ( G ) → { - 1 , 1 } is a signed dominating function (SDF) if the sum of its function values over any closed neighborhood is at least one. The weight of an SDF is the sum of its function values over all vertices. The signed domination number of G is the minimum weight of an SDF on G . In...

Bounds for the (Laplacian) spectral radius of graphs with parameter α

Gui-Xian Tian, Ting-Zhu Huang (2012)

Czechoslovak Mathematical Journal

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Let G be a simple connected graph of order n with degree sequence ( d 1 , d 2 , ... , d n ) . Denote ( α t ) i = ∑ j : i ∼ j d j α , ( α m ) i = ( α t ) i / d i α and ( α N ) i = ∑ j : i ∼ j ( α t ) j , where α is a real number. Denote by λ 1 ( G ) and μ 1 ( G ) the spectral radius of the adjacency matrix and the Laplacian matrix of G , respectively. In this paper, we present some upper and lower bounds of λ 1 ( G ) and μ 1 ( G ) in terms of ( α t ) i , ( α m ) i and ( α N ) i . Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.

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...