In the early 60s, Harary and Hill conjectured H(n) := 1 4b2 cbn−1 2 cbn−2 2 cbn−3 2 c to be the minimum number of crossings among all drawings of the complete graph Kn. It has recently been shown that this conjecture holds for so-called shellable drawings of Kn. For n ≥ 11 odd, we construct a non-shellable family of drawings of Kn with exactly H(n) crossings. In particular, every edge in our dr...

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) the Harary (also called matrix) while diag(RH(G)) represents diagonal of total vertices. In present work, some upper and lower bounds second-largest eigenvalue graphs in terms various parameters are investigated. Besides, all attaining these new characterized. Addi...

The stability of a communication network composed of processing nodes and communication links is of prime importance to network designers. As the network begins losing links or nodes, eventually there is a loss in its effectiveness. Thus, communication networks must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible...

The notion of resolving sets in a graph was introduced by Slater (1975) and Harary and Melter (1976) as a way of uniquely identifying every vertex in a graph. A set of vertices in a graph is a resolving set if for any pair of vertices x and y there is a vertex in the set which has distinct distances to x and y. A smallest resolving set in a graph is called a metric basis and its size, the metri...

Given an integer r > 0, let G r = (V; E) denote a graph consisting of a simple nite undirected connected nontrivial graph G together with r isolated vertices K r. Let L : V ! Z + denote a labelling of the vertices of G r with distinct positive integers. Then G r is said to be a sum graph if there exists a labelling L such that for every distinct vertex pair u and v of V , (u; v) 2 E if and only...

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 Kauffman-Harary conjecture is a conjecture for Fox’s colorings of alternating knots with prime determinants. We consider a conjecture for Alexander quandle colorings by referring to the Kauffman-Harary conjecture. We prove that this new conjecture is true for twist knots.

Abstract. Topological indices are the numerical value associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. Graph theory is a delightful playground for the exploration of proof techniques in Discrete Mathematics and its results have applications in many areas of sciences. A graph is a ...

Frank Harary in his article [2] was the first to speak of combinatorial diseases (actually, he spoke of graphical diseases). His original diseases were the four-colour disease, the hamiltonian disease and the reconstruction disease. Shortly thereafter, Read and Corneil [9] have discussed the graph isomorphism disease, and somewhat later, Huang, Kotzig and Rosa [4] spoke of a labelling disease. ...

Let G=(V; E) be a connected graph and S ⊂E. S is said to be a m-restricted edge cut (m-RC) if G − S is disconnected and each component contains at least m vertices. The m-restricted edge connectivity (G) is the minimum size of all m-RCs in G. Based on the fact that (G)6 3(G), where m(G)=min{!(X ): X ⊂V; |X |=m and G[X ] is connected} (!(X ) denotes the number of edges with one end vertex in X a...