Let G l and G 2 be (simple) graphs. The Ramsey*number r(Gl ,G 2) is the smallest integer n such that if one colors the complete graph Kn in two colors I and II, then either color I contains G 1 as a subgraph or color II contains G 2. The systematic study of r(Gl ,G2) was initiated by F. Harary, although there were a few previous scattered results of We also note here that notation not defined f...

This note addresses the following problem posed by S . M. Ulam. Find a concise expression for the number of classes of sets of n indistinguishable elements . Here, a class of sets is a set whose members are subsets of a given finite set of n elements, and referring to the elements as indistinguishable means that two classes are identical if some permutation on the elements transforms one class ...

A fuzzy graph is utilized to characterize the role played by an individual member in such a group that a class of group members having relationship with any given member has no sharply defined boundary. The concepts of weakening and strengthening points of an ordinary graph presented by Ross and Harary are generalized to those of a fuzzy graph.

A graph G is well-covered if all its maximal stable sets have the same size, denoted by α(G) (M. D. Plummer, 1970). If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G;x) = α(G) ∑ k=0 skx k is the independence polynomial of G (I. Gutman and F. Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured th...

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...

We show that the square of every connected S(K1,4)-free graph satisfying a matching condition has 2-connected spanning subgraph maximum degree at most 3. Furthermore, we characterise trees whose k. This generalises results on S(K1,3)-free graphs Henry and Vogler [7] Harary Schwenk [6], respectively.

We consider variants of the triangle-avoidance game first defined by Harary and rediscovered by Hajnal a few years later. A graph game begins with two players and an empty graph on n vertices. The two players take turns choosing edges within Kn, building up a simple graph. The edges must be chosen according to a set of restrictions R. The winner is the last player to choose an edge that does no...

In a paper of Harary and Plantholt, they concluded by noting that they knew of no generalization of the leaf edge exchange (LEE) transition sequence result on spanning trees to other natural families of spanning subgraphs. Now, we give two approaches for such a generalization. We define two kinds of LEE-graphs over the set of all connected spanning k-edge subgraphs of a connected graph G, and s...

A graph G=(V; E) is said to be an integral sum graph (sum graph) if its vertices can be given a labeling with distinct integers (positive integers), so that uv ∈ E if and only if u+ v ∈ V . The integral sum number (sum number) of a given graph G, denoted by (G) ( (G)), was de:ned as the smallest number of isolated vertices which when added to G result in an integral sum graph (sum graph). In th...