A graph is called radially maximal if it not complete and the addition of any new edge decreases its radius. In 1976 Harary Thomassen proved that radius $r$ diameter $d$ satisfy $r\le d\le 2r-2.$ Dutton, Medidi Brigham rediscovered this result with a different proof in 1995 they posed conjecture converse true, is, are positive integers satisfying 2r-2,$ then there exists $d.$ We prove little mo...

Abstract. The m, n Turk’s Head Knot, THK(m, n), is an “alternating (m, n) torus knot.” We prove the Harary-Kauffman conjecture for all THK(m, n) except for the case where m ≥ 5 is odd and n ≥ 3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observation that none of these knots have prime determinant except for THK(m, 2) when Pm ...

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.

Design: This cross-sectional, prospective, observational, multicenter study was performed in France, Germany, and Italy. Patients were stratified into 4 severity groups (BEVA, 20/40; WEVA, 20/200). Patients completed the National Eye Institute 25-Item Visual Function Questionnaire, the Macular Disease Quality of Life Scale, and the Health Utility Index 3. Analysis of variance was used to adjust...

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...