18 research outputs found
Parallelization of Modular Algorithms
In this paper we investigate the parallelization of two modular algorithms.
In fact, we consider the modular computation of Gr\"obner bases (resp. standard
bases) and the modular computation of the associated primes of a
zero-dimensional ideal and describe their parallel implementation in SINGULAR.
Our modular algorithms to solve problems over Q mainly consist of three parts,
solving the problem modulo p for several primes p, lifting the result to Q by
applying Chinese remainder resp. rational reconstruction, and a part of
verification. Arnold proved using the Hilbert function that the verification
part in the modular algorithm to compute Gr\"obner bases can be simplified for
homogeneous ideals (cf. \cite{A03}). The idea of the proof could easily be
adapted to the local case, i.e. for local orderings and not necessarily
homogeneous ideals, using the Hilbert-Samuel function (cf. \cite{Pf07}). In
this paper we prove the corresponding theorem for non-homogeneous ideals in
case of a global ordering.Comment: 16 page
Degree based Topological indices of Hanoi Graph
International audienceThere are various topological indices for example distance based topological indices and degree based topological indices etc. In QSAR/QSPR study, physiochemical properties and topological indices for example atom bond connectivity index, fourth atom bond connectivity index, Randic connectivity index, sum connectivity index, and so forth are used to characterize the chemical compound. In this paper we computed the edge version of atom bond connectivity index, fourth atom bond connectivity index, Randic connectivity index, sum connectivity index, geometric-arithmetic index and fifth geometric-arithmetic index of Double-wheel graph and Hanoi graph. The results are analyzed and the general formulas are derived for the above mentioned families of graphs
Eccentricity-Based Topological Invariants of Some Chemical Graphs
Topological index is an invariant of molecular graphs which correlates the structure with different physical and chemical invariants of the compound like boiling point, chemical reactivity, stability, Kovatâs constant etc. Eccentricity-based topological indices, like eccentric connectivity index, connective eccentric index, first Zagreb eccentricity index, and second Zagreb eccentricity index were analyzed and computed for families of Dutch windmill graphs and circulant graphs
First and Second Zagreb Eccentricity Indices of Thorny Graphs
The Zagreb eccentricity indices are the eccentricity reformulation of the Zagreb indices. Let H be a simple graph. The first Zagreb eccentricity index ( E 1 ( H ) ) is defined to be the summation of squares of the eccentricity of vertices, i.e., E 1 ( H ) = â u â V ( H ) Ć H 2 ( u ) . The second Zagreb eccentricity index ( E 2 ( H ) ) is the summation of product of the eccentricities of the adjacent vertices, i.e., E 2 ( H ) = â u v â E ( H ) Ć H ( u ) Ć H ( v ) . We obtain the thorny graph of a graph H by attaching thorns i.e., vertices of degree one to every vertex of H . In this paper, we will find closed formulation for the first Zagreb eccentricity index and second Zagreb eccentricity index of different well known classes of thorny graphs
Topological Indices of H-Naphtalenic Nanosheet
In chemical graph theory, a single numeric number related to a chemical structure is called a topological descriptor or topological index of a graph. In this paper, we compute analytically certain topological indices for H-Naphtalenic nanosheet like Randic index, first Zagreb index, second Zagreb index, geometric arithmetic index, atom bond connectivity index, sum connectivity index and hyper-Zagreb index using edge partition technique. The first multiple Zagreb index and the second multiple Zagreb index of the nanosheet are also discussed in this paper
On Trees with Given Independence Numbers with Maximum Gourava Indices
In mathematical chemistry, molecular descriptors serve an important role, primarily in quantitative structureâproperty relationship (QSPR) and quantitative structureâactivity relationship (QSAR) studies. A topological index of a molecular graph is a real number that is invariant under graph isomorphism conditions and provides information about its size, symmetry, degree of branching, and cyclicity. For any graph N, the first and second Gourava indices are defined as GO1(N)=âuâ˛vâ˛âE(N)(d(uâ˛)+d(vâ˛)+d(uâ˛)d(vâ˛)) and GO2(N)=âuâ˛vâ˛âE(N)(d(uâ˛)+d(vâ˛))d(uâ˛)d(vâ˛), respectively.The independence number of a graph N, being the cardinality of its maximal independent set, plays a vital role in reading the energies of chemical trees. In this research paper, it is shown that among the family of trees of order δ and independence number Ξ, the spur tree denoted as μδ,Ξ maximizes both 1st and 2nd Gourava indices, and with these characterizations this graph is unique