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Strong Equality Between the Roman Domination and Independent Roman Domination Numbers in Trees

Mustapha Chellali, Nader Jafari Rad (2013)

Discussiones Mathematicae Graph Theory

A Roman dominating function (RDF) on a graph G = (V,E) is a function f : V −→ {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF is the value f(V (G)) = P u2V (G) f(u). An RDF f in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number R(G) (respectively, the independent Roman domination number iR(G)) is the minimum weight of an RDF (respectively,...

Structure of cubic mapping graphs for the ring of Gaussian integers modulo n

Yangjiang Wei, Jizhu Nan, Gaohua Tang (2012)

Czechoslovak Mathematical Journal

Let ℤ n [ i ] be the ring of Gaussian integers modulo n . We construct for ℤ n [ i ] a cubic mapping graph Γ ( n ) whose vertex set is all the elements of ℤ n [ i ] and for which there is a directed edge from a ∈ ℤ n [ i ] to b ∈ ℤ n [ i ] if b = a 3 . This article investigates in detail the structure of Γ ( n ) . We give suffcient and necessary conditions for the existence of cycles with length t . The number of t -cycles in Γ 1 ( n ) is obtained and we also examine when a vertex lies on a t -cycle of Γ 2 ( n ) , where Γ 1 ( n ) is induced by all the units of ℤ n [ i ] while Γ 2 ( n ) is induced by all the...

Structures ofW(2.2) Lie conformal algebra

Lamei Yuan, Henan Wu (2016)

Open Mathematics

The purpose of this paper is to study W(2, 2) Lie conformal algebra, which has a free ℂ[∂]-basis L, M such that [...] [LλL]=(∂+2λ)L,[LλM]=(∂+2λ)M,[MλM]=0 . In this paper, we study conformal derivations, central extensions and conformal modules for this Lie conformal algebra. Also, we compute the cohomology of this Lie conformal algebra with coefficients in its modules. In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.

Subarborians

Bohdan Zelinka (1980)

Czechoslovak Mathematical Journal

Survey of certain valuations of graphs

Martin Bača, J.A. MacDougall, Mirka Miller, Slamin, W.D. Wallis (2000)

Discussiones Mathematicae Graph Theory

The study of valuations of graphs is a relatively young part of graph theory. In this article we survey what is known about certain graph valuations, that is, labeling methods: antimagic labelings, edge-magic total labelings and vertex-magic total labelings.

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