An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. If sk is the number of independent sets of cardinality k in G, then I(G;x) = s0 + s1x+ s2x 2 + ...+ sαx α, α = α (G) , is called the independence polynomial of G (I. Gutman and F. Harary, 1983). If sα−i = f (i) · si holds for every i ∈ {0, 1, ..., ⌊α/2⌋}, t...

We introduce a new family of graphs for which the Hamiltonian path problem is non-trivial and yet has a linear time solution. The square of a graph G = (V, E), denoted as G, is a graph with the set of vertices V , in which two vertices are connected by an edge if there exists a path of length at most 2 connecting them in G. Harary & Schwenk (1971) proved that the square of a tree T contains a H...

abstract: in this study an appropriate computational approach was presented for estimating the boiling temperatures of 41 different types of olefins and their derivatives. based on the guidelines of this approach, several structural indices related to the organic components were applied using graph theory. meanwhile, in addition to evaluating the relation between the boiling temperatures of ole...

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. one of the us...

Most analyses of world politics and studies of national security policies recognize the interdependence among the salient actors across the salient issues. Traditionally, international politics has been defined as the scope and extent of the relations among independent countries, thought to be the most important elements in world politics. This means that actors as well as their actions are str...

With the help of computer algorithms, we improve the lower bound on the Ramsey multiplicity of K4, and thus show that the exact value of it is equal to 9. The Ramsey multiplicity M(G) of a graph G is defined as the smallest number of monochromatic copies ofG in any two-coloring of edges of KR(G), whereR(G) is the Ramsey number of G, i.e. the smallest integer n such that any two-coloring of edge...

The line graph L(G) of a graph G is defined to have as its vertices the edges of G, with two being adjacent if the corresponding edges share a vertex in G. Line graphs have a rich history. The name line graph was first used by Harary and Norman in 1960. But line graphs were the subject of investigation as far back as 1932 in a paper by H. Whitney in which he showed that for connected graphs, ed...

A graph G is called integral if all zeros of the characteristic polynomial P (G, x) are integers. Our purpose is to determine or to characterize which graphs are integral. This problem was posed by Harary and Schwenk in 1974. In general, the problem of characterizing integral graphs seems to be very difficult. Thus, it makes sense to restrict our investigations to some interesting families of g...

An eulerian subgraph of a graph is called a circuit. As shown by Harary and Nash-Williams, the existence of a Hamilton cycle in the line graph L(G) of a graph G is equivalent to the existence of a dominating circuit in G, i.e., a circuit such that every edge of G is incident with a vertex of the circuit. Important progress in the study of the existence of spanning and dominating circuits was ma...

Assuming that G is a finite group and H subgroup of G, the graph known as relative coprime with respect to (denoted Γ_(G,H)) has vertices corresponding elements G. Two distinct x y are adjacent by an edge if only (|x|,|y|)=1 or belongs H. This paper will focus on finding general formula for some topological indices dihedral group. The study in theory offers valuable insights into structural pro...