On k-Equivalence Domination in Graphs

Let G = (V,E) be a graph. A subset S of V is called an equivalence set if every component of the induced subgraph (S) is complete. If further at least one component of (V − S) is not complete, then S is called a Smarandachely equivalence set

The equivalence chain of a graph

Let <i>G</i> = (<i>V, E</i>) be a graph. A subset <i>S</i> of <i>V</i> is called an <i>equivalence set</i> if every component of the induced subgraph <<i>S</i> is complete. In this paper starting with the concept of equivalence set as seed property, we form an inequality chain of six parameters, which we call the <i>equivalence chain</i> of <i>G</i>. WE present several basic results on these parameters and problems for further investigation

On equality in an upper bound for the equivalence domination number

Let G = (V, E) be a graph. A subset S of V is called an equivalence set if every component of the induced subgraph ⟨S⟩ is complete. The equivalence domination number γe(G) is the minimum cardinality of an equivalence dominating set of G. In this paper we investigate the structure of graphs G satisfying γe(G) = |V (G)| − Δ(G).Keywords: Dominating set, equivalence set, independent set, equivalence domination number

Local edge coloring of graphs

Let be a graph. A local edge coloring of G is a proper edge coloring such that for each subset S of E(G) with there exist edges such that where ns is the number of copies of P3 in the edge induced subgraph The maximum color assigned by a local edge coloring c to an edge of G is called the value of c and is denoted by The local edge chromatic number of G is where the minimum is taken over all local edge colorings c of G. In this article, we derive bounds and many results based on local edge coloring

Equivalence dominating sets in graphs

Let G = (V,E) be a graph. A subset S of V is called an equivalence set if every component of the induced subgraph (S) is complete. In this paper we introduce several parameters using equivalence sets and discuss their relation with other graph theoretic parameters

Phonon vibrational frequencies and elastic properties of solid SrFCl. An ab initio study

The phonon vibrational frequencies, electronic and elastic properties of SrFCl, one of the members of the alkaline-earth fluorohalide family crystallizing with the PbFCl-type structure, have been investigated, for the first time, at the ab initio level, by using the periodic CRYSTAL program. Both Hartree-Fock (HF) and density functional theory (DFT) Hamiltonians have been used, with the latter in its local density, gradient-corrected (PW91), and hybrid (B3LYP) versions. The structural and elastic properties are in good agreement with experiment, with the exception of those calculated within the local density approximation, which were found to be systematically under-estimated (distances) or over-estimated (elastic properties). As regards the phonon frequencies, B3LYP and PW91 provide excellent results, the mean absolute difference with respect to the experimental Raman data being 4.1% and 3.6%, respectively. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2005