Let G = (V,E) be a graph. A k-coloring of a graph G is a labeling f : V (G) → T , where | T |= k and it is proper if the adjacent vertices have different labels. A graph is k-colorable if it has a proper k-coloring. The chromatic number χ(G) is the least k such that G is k-colorable. Here we study chromatic numbers in some kinds of Harary graphs. Mathematics Subject Classification: 05C15

The metric dimension problem was first introduced in 1975 by Slater [12], and independently by Harary and Melter [6] in 1976; however the problem for hypercube was studied (and solved asymptotically) much earlier in 1963 by Erdős and Rényi [4]. A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G...

J. A. Bondy and U. S. R. Murty, Graph theory with Applications, London: MacMillan (1976). R. M. Gera, On Dominator Colorings in Graphs, Graph Theory Notes of New York LIT, 25-30 (2007) F. Harary, Graph Theory, Narosa Publishing 1969. D. Michalak, On Middle and Total graphs with Coarseness Number Equal 1, Lecture Notes in Mathematics, Volume 1018: Graph Theory, Springer-Verlag, Berlin, 139 – 150...

The Harary-Hill conjecture states that the minimum number of crossings in a drawing of the complete graph Kn is Z(n) := 1 4 ⌊ n 2 ⌋ ⌊ n−1 2 ⌋ ⌊ n−2 2 ⌋ ⌊ n−3 2 ⌋ . This conjecture was recently proved for 2-page book drawings of Kn. As an extension of this technique, we prove the conjecture for monotone drawings of Kn, that is, drawings where all vertices have different x-coordinates and the edg...

Sonlu bir Z_n grubunun yönlendirilmemiş P(Z_n ) power grafı, köşelerinin kümesi olan bağlantılı graftır ve burada iki köşe komşudur gerek yeter şart u≠v ⊆ veya dir. 〖(Z〗_n,+) grubuna karşılık gelen Yönlendirilmemiş G=P(Z_n grafının Hyper-Wiener, SK, 〖SK〗_1, 〖SK〗_2 Harary indeksleri sırasıyla, 1/2 ∑_({u,v}⊆V(G))▒〖d(u,v)〗+1/2 ∑_({u,v}⊆V(G))▒〖d^2 (u,v) 〗 ∑_(uv∈E(G))▒〖d(u)+d(v)〗, ∑_(uv∈E(G))▒〖d(u)....

and Dedication This paper is an exploration of simple four-regular graphs in the plane (i.e. loopless and with no more than one edge between any two nodes). Such graphs are fundamental to the theory of knots and links in three dimensional space, and their planar diagrams. We dedicate this paper to Frank Harary (1921 – 2005) whose fascination with graphs of knots inspired this work and with whom...

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 ...