It is well known that the chemical behavior of a compound is dependent upon the structure of itsmolecules. Quantitative structure – activity relationship (QSAR) studies and quantitative structure –property relationship (QSPR) studies are active areas of chemical research that focus on the nature ofthis dependency. Topological indices are the numerical value associated with chemical constitution...

A linear forest is the union of a set of vertex disjoint paths. Akiyama, Exoo and Harary and independently Hilton have conjectured that the edges of every graph of maximum degree ∆ can be covered by d∆+1 2 e linear forests. We show that almost every graph can be covered with this number of linear forests. 1 Linear Forests, Directed Cycles, and the Probabilistic Method A linear forest is the uni...

Let G1 and G2 be disjoint copies of a graph G, and let f : V (G1) → V (G2) be a function. Then a functigraph C(G, f) = (V,E) has the vertex set V = V (G1) ∪ V (G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | u ∈ V (G1), v ∈ V (G2), v = f(u)}. A functigraph is a generalization of a permutation graph (also known as a generalized prism) in the sense of Chartrand and Harary. In this paper, we study ...

Graf koprima pada grup bilangan bulat modulo memiliki banyak aplikasi penting dalam matematika diskrit dan kriptografi. Indeks Harary dari graf sangat berguna ilmu komputer, terutama perhitungan properti aljabarik, topologi, struktur graf. Pada artikel ini didapatkan Hararay untuk yang berorde perpangkatan prima, yakni kuadrat jumlah anggota dikurangi satu.

The concept of continuous monotonic decomposition (CMD) was INTRODUCTION By a graph we mean a finite undirected graph without loops or multiple edges. Terms not defined here are used in the sense of Harary [1].

The terminal Steiner tree problem is a special version of the Steiner tree problem, where a Steiner minimum tree has to be found in which all terminals are leaves. We prove that no polynomial time approximation algorithm for the terminal Steiner tree problem can achieve an approximation ratio less than (1− o(1)) lnn unless NP has slightly superpolynomial time algorithms. Moreover, we present a ...

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs. We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing K4 or K2,3 as a minor is Hamiltonian if and only if it is 2-connected. We show furth...

The aim of this note is to establish new spectral bounds for the harmonic matrix. The Harary matrix of given a connected graph G of order n, say RD(G), is an n-by-n symmetric matrix, such that (RD(G))ij =  1 dij , if i < j, 0 , if i = j, where dij denotes the distance between the vertices i and j [10, 11]. This matrix (originally known as reciprocal distance matrix [11]) is particulary well-k...

Given two sets of points in the plane, P of n terminals and S of m Steiner points, a Steiner tree of P is a tree spanning all points of P and some (or none or all) points of S. A Steiner tree with length of longest edge minimized is called a bottleneck Steiner tree. In this paper, we study the Euclidean bottleneck Steiner tree problem: given two sets, P and S, and a positive integer k ≤ m, find...

The rectilinear Steiner tree problem is to nd a minimum-length rectilinear interconnection of a set of points in the plane. A reduction from the rectilinear Steiner tree problem to the graph Steiner tree problem allows the use of exact algorithms for the graph Steiner tree problem to solve the rectilinear problem. Furthermore, a number of more direct, geometric algorithms have been devised for ...