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...

The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph Kn in the plane is at least Z(n) := 1 4 ⌊ n 2 ⌋ ⌊ n−1 2 ⌋ ⌊ n−2 2 ⌋ ⌊ n−3 2 ⌋ . In this paper, we settle the Harary-Hill conjecture for shellable drawings. We say that a drawing D of Kn is s-shellable if there exist a subset S = {v1, v2, . . . , vs} of the vertices and a region R of D with the fo...

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...

Introduced in 1947, the Wiener index W (T ) = ∑ {u,v}⊆V (T ) d(u, v) is one of the most thoroughly studied chemical indices. The extremal structures (in particular, trees with various constraints) that maximize or minimize the Wiener index have been extensively investigated. The Harary index H(T ) = ∑ {u,v}⊆V (T ) 1 d(u,v) , introduced in 1993, can be considered as the “reciprocal analogue” of ...

The acyclic point-connectivity of a graph G, denoted (G), is the minimum number of points whose removal from G results in an acyclic graph. In a 1975 paper, Harary stated erroneously that (Qn 2 n-I i where Qn denotes the n-cube. We prove that for n > 4, 7 2 n-4 -< () -< 2 n-I 2n-y-2, where y [log2(ni)]. We show that the upper bound is obtained for n -< 8 and conjecture that it is obtained for a...

We investigate families of two-dimensional simplicial complexes defined in terms of vertex decompositions. They include nonevasive complexes, strongly collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary and Palmer. We investigate the complexity of recognition problems for those families and some of their combinatorial properties. Certain results follow from analogous d...

A (k; g)-cage is a k-regular graph of girth g with minimum order. In this work, for all k ≥ 2 and g ≥ 5 odd, we present an upper bound of the order of a (k; g + 1)-cage, which depends on the order of a (k; g)-cage, improving a previous result of Sauer of 1967. We also show that every (k; 11)-cage contains a cycle of length 12, confirming a case of a conjecture of Harary and Kóvacs of 1983.

The interval number of a graph G, denoted i(G), is the least positive integer t for which G is the intersection graph of a family of sets each of which is the union of at most t cIosed intervals of the real line IR. Trotter and Harary showed that the interval number of the complete bipartite graph K(m, n) is [(mn + I)/(m + n)]. Matthews showed that the interval number of the complete multiparti...

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...

A graph G has a representation modulo r if there exists an injective map f : V (G) → {0, 1, ..., r − 1} such that vertices u and v are adjacent if and only if f(u) − f(v) is relatively prime to r. The representation number of G, rep(G), is the smallest integer such that G has a representation modulo r. In this paper we study the representation numbers of various graphs, such as the complete ter...