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Trees with equal total domination and total restrained domination numbers

Xue-Gang ChenWai Chee ShiuHong-Yu Chen — 2008

Discussiones Mathematicae Graph Theory

For a graph G = (V,E), a set S ⊆ V(G) is a total dominating set if it is dominating and both ⟨S⟩ has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S ⊆ V(G) is a total restrained dominating set if it is total dominating and ⟨V(G)-S⟩ has no isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. We characterize all trees for which total domination and total restrained...

The Laplacian spectral radius of graphs

Jianxi LiWai Chee ShiuAn Chang — 2010

Czechoslovak Mathematical Journal

The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we improve Shi's upper bound for the Laplacian spectral radius of irregular graphs and present some new bounds for the Laplacian spectral radius of some classes of graphs.

On the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph

Ji-Ming GuoJianxi LiWai Chee Shiu — 2013

Czechoslovak Mathematical Journal

The Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph are the characteristic polynomials of its Laplacian matrix, signless Laplacian matrix and normalized Laplacian matrix, respectively. In this paper, we mainly derive six reduction procedures on the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph which can be used to construct larger Laplacian, signless Laplacian and normalized Laplacian cospectral graphs, respectively....

On the total restrained domination number of direct products of graphs

Wai Chee ShiuHong-Yu ChenXue-Gang ChenPak Kiu Sun — 2012

Discussiones Mathematicae Graph Theory

Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V∖S is adjacent to a vertex in S as well as to another vertex in V∖S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by γ r t ( G ) , is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that these bounds...

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