It is shown that every n vertex triple system with every pair of vertices lying in at least (1/2 + o(1))n triples contains a copy of the Fano plane. The constant 1/2 is sharp. This is the first triple system for which such a (nonzero) constant is determined. Let X be a finite set. An r-graph F with vertex set X is a family of r-element subsets of X. These subsets are called edges, and X is writ...

the first zagreb index $m_1$ of a graph $g$ is equal to the sum of squaresof degrees of the vertices of $g$. goubko proved that for trees with $n_1$pendent vertices, $m_1 geq 9,n_1-16$. we show how this result can beextended to hold for any connected graph with cyclomatic number $gamma geq 0$.in addition, graphs with $n$ vertices, $n_1$ pendent vertices, cyclomaticnumber $gamma$, and minimal $m...

the crossing number of a graph  is the minimum number of edge crossings over all possible drawings of  in a plane. the crossing number is an important measure of the non-planarity of a graph, with applications in discrete and computational geometry and vlsi circuit design. in this paper we introduce vertex centered crossing number and study the same for maximal planar graph.

Large scale graph processing is a major research area for Big Data exploration. Vertex centric programming models like Pregel are gaining traction due to their simple abstraction that allows for scalable execution on distributed systems naturally. However, there are limitations to this approach which cause vertex centric algorithms to under-perform due to poor compute to communication overhead ...

Abstract Let R be a finite commutative ring. We define co-unit graph, associated to ring , denoted by G nu ( ) with vertex set V )) = U ), where is the of units and two distinct vertices x y being adjacent if only + ∉ / ). In this paper, we investigate some basic properties integers modulo n, for different values n. find domination number, clique number girth

in this paper, we present some inequalities for the co-pi index involving the some topological indices, the number of vertices and edges, and the maximum degree. after that, we give a result for trees. in addition, we give some inequalities for the largest eigenvalue of the co-pi matrix of g.

in this paper, we introduce the notion of primal strong co-ideals and give some results involving them. it is shown thatsubtractive strong co-ideals are intersection of all primal strong co-ideals that contain them. also we prove that the representation of strong co-ideals as reduced intersections of primal strong co-ideals is unique.

The neutral ferratricarbaboranes (η(5)-RC3B8H10)Fe(η(5)-C5H5) (R = aryl), analogues of ferrocene, have recently (2013) been synthesized by Štíbr and co-workers and structurally characterized by X-ray crystallography. We have now used density functional theory to study the structures of the complete series of related ferratricarbaboranes CpFeC3B(n-4)H(n-1) (n = 8 to 12). The lowest energy struct...