We prove in a new way that the maximum number of edges an induced subgraph of the 2-grid Z on n vertices is given by E2(n) = ⌊2n − √ 2n⌋. This was first proved by Harary and Harborth in 1976 by induction. Our method suggests some possible ways to obtain analog results in higher dimensions via projections and the Box Theorem by Bollobás and Thomas from 1995. – We discuss some of these key tools ...

The eccentricity of a vertex is the maximum distance from it to another vertex, and the average eccentricity of a graph is the mean eccentricity of a vertex. In this paper we introduce average edge and average vertex-edge mean eccentricities of a graph. Moreover, relations among these eccentricities for trees are provided as well as formulas for line graphs and cartesian product of graphs. In t...

A tripartite-circle drawing of a tripartite graph is in the plane, where each part vertex partition placed on one three disjoint circles, and edges do not cross circles. We present upper lower bounds minimum number crossings drawings $K_{m,n,p}$. In contrast to 1- 2-circle drawings, which may attain Harary-Hill bound, our results imply that balanced restricted 3-circle complete are optimal.

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

Let G and H be simple graphs. The Ramsey number r(G,H) for a pair of graphs G and H is the smallest number r such that any red-blue coloring of the edges of Kr contains a red subgraph G or a blue subgraph H . The size Ramsey number r̂(G,H) for a pair of graphs G and H is the smallest number r̂ such that there exists a graph F with size r̂ satisfying the property that any red-blue coloring of the e...

Jochen Reibenwein, Dietmar Pils, Peter Horak, Birgit Tomicek, Gregor Goldner, Nina Worel, Katarzyna Elandt, and Michael Krainer* DivisionofOncology,Departmentof InternalMedicine I,Medical Universityof Vienna,Vienna, Austria Departmentof RadiotherapyandRadiobiology,Medical Universityof Vienna,Vienna, Austria Departmentof Bloodgroup SerologyandTransfusionMedicine,Medical Universityof Vienna,Vienn...

If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G;x) = α(G) ∑ k=0 skx k is the independence polynomial of G (Gutman and Harary, 1983). A graph G is very well-covered (Favaron, 1982) if it has no isolated vertices, its order equals 2α(G) and it is well-covered (i.e., all its maximal independent sets are of the same size, M...

The central problems in social sciences concern the social and psychological mechanisms and conditions required for the emergence and stability of human groups. The present article is dedicated to the problem of stability of human groups. We model human groups using local interacting systems of automaton with relations and reactions and using the structural balance theory. The ‘structural balan...

In a linear forest, each component is a path. The linear arboricity ~(G) of a graph G is defined in Harary [8] as the minimum number of linear forests whose union is G. This invariant first arose in a study [i0] of information retrieval in file systems. A quite similar covering invariant which is well known to the linear arboricity is the arboricity of a graph, which is defined as the minimum n...

If G and H are graphs, define the Ramsey number to be the least number p such that if the lines of the complete graph Kp are colored red and blue (say),either the red subgraph contains a copy of G or the blue subgraph contains H. Also set r(G) = r(G,G) ; these are called the diagonal Ramsey numbers. These definitions are taken from Chvátal and Harary [1J ; other terminology will follow Harary [...