The Harary-Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph Kn is H(n) = 1 4 ⌊ n 2 ⌋⌊ n− 1 2 ⌋⌊ n− 2 2 ⌋⌊ n− 3 2 ⌋ . Ábrego et al. [3] introduced the notion of shellability of a drawing D of Kn. They proved that if D is s-shellable for some s ≥ b 2 c, then D has at least H(n) crossings. This is the first combinatorial condition on a dr...

In an effort to continue the pioneering work of Harary in USA and Flament in France, we have undertaken to develop, on an experimental basis, a formalized theory of systems of beliefs and their modifications. This theory uses the psycho-social concepts of theories of cognitive consistency and of the tools of discrete mathematics, such as rewriting and intervals within graphs. The axioms and rew...

A linear forest-factor F of a graph G is a spanning subgraph of G whose components are paths. A linear forest-decomposition of G is a collection :F = {F1, ••• , Fk } of linear forest-factors of G such that the edge set E (G) of G is the disjoint union of E (F1), •.• , E (Fk)' The linear ar borici ty la( G) of G is the minimum cardinality of a linear forest-decomposition of G. In this paper we e...

The geodetic problem was introduced by Harary et al. [17]. In order to model some social network problems, a similar problem is introduced in this paper and named the strong geodetic problem. The problem is solved for complete Apollonian networks. It is also proved that in general the strong geodetic problem is NP-complete.

The relationship between the Randic , Wiener, Hosoya , Balaban, Schultz indices, Harary numbers andDistance matrix to enthalpies of formation (Airf), heat capacity, (Cp) , enthalpies of combustion (AH °c ),enthalpy of vaporization (AH °vap) and normal boiling points (bpK)of C2 C10 normal alkanes isrepresented

We give a complete answer to a question raised by Harary and Manvel in 1972 [3] by proving that a finite set A of points in the plane R is uniquely determined up to translation and rotation by a multiple of 90◦ by 5 of its (|A| − 1)-element subsets given up to translation and rotation by a multiple of 90◦. This result is best-possible.

We study 0-1 laws for extensions of rst-order logic by Lindstrr om quantiiers. We state suucient conditions on a quantiier Q expressing a graph property, for the logic FOQ] { the extension of rst-order logic by means of the quantiier Q { to have a 0-1 law. We use these conditions to show, in particular, that FORig], where Rig is the quantiier expressing rigidity, has a 0-1 law. We also show tha...

Harary et al. put forward the concept on minimum cardinality over all subsets of perfect matching M that are not included by any other ones, to be forcing number for M. A counting polynomial matchings possessing same was introduced Zhang al., using name `forcing polynomial'. This research deduces recurrence formulas polynomials monotonic constructable hexagonal systems and with one turning. Fro...

A linear forest is a in which every connected component path. The arboricity of graph G the minimum number forests covering all edges. In 1980, Akiyama, Exoo, and Harary proposed conjecture, known as Linear Arboricity Conjecture (LAC), stating that Δ-regular has ⌈ Δ + 1 2 ⌉ . 1988, Alon proved LAC holds asymptotically. 1999, list version was raised by An Wu, called List Conjecture. this article...